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]]>For any scientific studies, designing and evolving suitable methodology of an experiment in an institution or for an inquiry in the field, need very logical and systematic planning. The knowledge of how experiments come in use and in what way it is to be carried out is discussed below.
Thus it is important for any researcher to know how to carry out an experiment.
An experiment deliberately imposes a treatment on a group of objects in the interest of observing the response.
The experimental unit is the physical entity that can be assigned, at random, to a treatment. It is also the unit of statistical analysis. Adequate replications are done after which the researcher makes an inference.
Treatment is something that researchers administer to experiment units. It is also referred to as an independent variable.
We are concerned with the analysis of data generated from an experiment. It is wise to take time and effort to organize the experiment properly to ensure that the right type of data and enough of it is available to answer the questions of interest as clearly and efficiently as possible. This process is called experimental design.
We design an experiment to improve the precision of our answers. Thus the specific questions that the experiment is intended to answer must be clearly identified before experimenting. We should also attempt to identify known or expected sources of variability in the experimental units since one of the main aims of a designed experiment is to reduce the effect of these sources of variability on the answers to questions of interest.
Before carrying out an experiment, we should know what kinds and how many variables we are dealing with. Otherwise, the result will be less precise and may encounter an unexpected anomaly. There are three types of variables:
Other factors such as the nature of the study whether it is descriptive, analytical, or interventional could be identified to use the proper method of sampling and experiment. Observer and the instrumental error should also be taken into account while experimenting.
Reference
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]]>The post Comparison of Two Means: Student’s ttest appeared first on Plantlet.
]]>From the previous topic, we learned about test of significance, a procedure for comparing observed data with a claim or hypothesis or predicted data.
Student’s ttest
A ttest is a statistical test that is used to compare the means of two groups. It is often used in hypothesis testing to determine whether a process or treatment actually has an effect on the population of interest, or whether two groups are different from one another.
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Formula of student’s ttest (and Z test)
A. Estimating the difference of means of two samples
B. Estimating the Standard error of that difference
Formula to estimate SE
Steps for student’s ttest
Step 1: Calculate the tvalue
Discussed earlier.
Step 2: Calculate the degrees of freedom
Step 3: Determine the critical value
N.B: If there’s no specified alpha level, we should use P0.05 (95% confidence level) and we must take the value of two tails.
Step 4: Compare the tstatistic value to critical value
So, the null hypothesis would be rejected and an alternative hypothesis should be taken. In this case, the estimates falls in the area of rejection.
Unpaired Samples
Example
Index of data of 40 boys and 46 girls gave following values.
Gender  Mean  SD^{2} 
Boys  53.11  228.57 
Girls  48.63  395.85 
From t table, we can find that the critical t value at 95% confidence level is 1.98 which is greater than our hypothesis value, hence the difference is probably due to chance that means not significant and also the null hypothesis is accepted.
2. Two small samples t test:
Example
Can we say that vitamins A and D were responsible for this difference?
Group A  Group B  
X_{A}  X_{A}^{2}  X_{B}  X_{B}^{2} 
5
3 4 3 2 6 3 2 3 6 7 5 3 
25
9 16 9 4 36 9 4 9 36 49 25 9 
1
3 2 4 2 1 3 4 3 2 2 3

1
9 4 16 4 1 9 16 9 4 4 9

Comments
df = 23
At 23 df the highest obtainable value of ‘t’ at 5% level of significance is 2.069 as found on reference to ‘t’ table. The ‘t’ value in this experiment is found at 2.74 which is much higher than the critical value. The significance is real and null hypothesis rejected.
Paired samples
Example
Plant height in cm is given below:
Serial no.  Height after exposed to light (cm)  Height after exposed to dark (cm)  Difference (cm) 
Difference^{square} 
X_{1}  X_{2}  x = X_{1 }– X_{2}  x^{2}  
1.  142  138  4  16 
2.  140  136  4  16 
3.  144  147  3  9 
4.  144  139  5  25 
5.  142  143  1  1 
6.  146  141  5  25 
7.  149  143  6  36 
8.  150  145  5  25 
9.  142  136  6  36 
10.  148  146  2  4 
Σ x = 33  Σ x^{2} = 193 
Comment
Reference
Revised by
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]]>The post Interrelationships of quantitative variables: Correlation and Regression appeared first on Plantlet.
]]>In the experiment, we measure the two continuous characters which are associated with each other. For example, the height of the plant and the temperature of the atmosphere. Accordingly, to understand the relationship between two such variables, we need to know how they are related and how the relationship can be expressed in a visual form.
The statistical technique to determine the relationship or association between two quantitative variables is called correlation. In other words, it determines the relationship between two quantitative variables. However, it does not prove that one particular variable causes the change in the other.
In correlation coefficient, we measure the degree of the relationship between two sets of figures in terms of another parameter. A simple correlation coefficient is denoted by the letter “r”. In addition, it is known as Pearson’s correlation or productmoment correlation coefficient. For population coefficient, we use the Greek letter “ρ”. Its pronunciation is “rho”. Certainly, the absolute value of r remains constant irrespective of change of origin.
The extent of correlation varies between minus one and plus one (1 ≤ r ≤ 1). The value is in a fraction with a positive or negative sign.
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It is common to use a scatter diagram as a visual representation of data. We can show on a graph paper by plotting each pair of variables (X and Y). Consequently, placing a dot at the point corresponding to the values of X and Y.
The two variables are moderately proportional to one another.
This calculation is introduced by Professor Karl Pearson. It is used to determine the direction and degree of the linear relationship between two variables. The variables must be normally distributed for this method to be applied.
Formula:
Where numerator indicates variability between two variables.
As we carry out an experiment and take observations from the sample, the observed value of “r” has to be tested for significance. The following formula is for the calculation of a small sample. We set up nullhypothesis as:
H^{o }= There is no significant relationship between dependent and independent variables.
Formula:
where the degrees of freedom = n2.
We use regression analysis to describe the relationships between a set of independent variables and the dependent variable. Regression analysis produces a regression equation. Moreover, the coefficients represent the relationship between each independent variable and the dependent variable. Therefore, it enables the user to predict the values of one variable on the basis of the other variable. For instance, on the positive or negative side, beyond the mean. Francis Galton coined the term “regression” in the nineteenth century. He described a biological phenomenon through regression.
The regression coefficient is denoted by the letter “b”. It shows the gradient or slope of the straight line of correlation. Moreover, it can calculate the equation for a straight line in correlation (Y= a+bX).
Formula:
The calculated b is placed into the equation. Consequently, “a” is a constant for Yintercept. We can find by subtracting the product of regression coefficient and mean of X from the mean of Y:
Finally, we plot a straight line. Place the value of X and Y. The lines will go through the points.
Reference:
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]]>The post Analysis of Variance and F test appeared first on Plantlet.
]]>We can test the significance of the two sample means by using a ttest. However, in cases where there are more than two samples, it is done by Analysis of Variance (ANOVA). This method is carried out by comparing sample variances using Variance Ratio Test or F test. So at first, let’s learn about the F test which is discussed below:
The variance ratio test was first conducted and named after a British statistician called Ronald Aylmer Fisher (17 February 1890 – 29 July 1962). It is any statistical test in which the test statistic has an Fdistribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled. The value of F is equal to the ratio of variance.
F= Estimate variance between groups ( including treatment effects)/ Estimate variance within groups (excluding treatment effects)
In ANOVA the F test is determined as the final goal that indicates the presence or absence of a significant difference between more than two samples. In the context of ANOVA, the sample variances are called mean squares, or MS values.
General logic and basic formulas for the hypothesis testing procedure known as analysis of variance (ANOVA). The purpose of ANOVA is much the same as the ttests. The goal is to determine whether the mean differences that are obtained for sample data are sufficiently large to justify the conclusion that there are meaningful differences between the populations from which the samples were obtained.
There are two methods of ANOVA depending on the condition of samples in which they are grouped.
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If the analysis concerns with testing the difference between two sets of groups without any further categorization according to the other factors concerned, we use oneway ANOVA test. Steps of finding F value in ANOVA is given with examples
Example The table is given where it shows a record of the number of new leaves observed under different concentrations of hormone given.
Step1: Write null hypothesis. For example here, it can be stated as “there is no significant difference/ variation among the treatments (different concentration of hormones)”
Step2: Calculate sum of square (Total SS) using the formula ∑x^{2 }− (∑x) ^{2 }⁄ n [where (∑x) ^{2 }⁄ n is the correction factor and “n” is the number of observed samples]
here, Total SS = (3^{2}+ 4 ^{2}+ 8^{2} +9^{2})− (7+17)^{2}/4
= 170− 144
= 26.
Step3: Calculate Treatment Sum of square using formula (total of 1st treatment)^{2}/n + …..+(total of last treatment)^{2}/n − (∑x) ^{2 }⁄ n [where n is the number of observation under one treatment]
here, Treatment SS = (7^{2}/2 + 17^{2}/2) − (7+17) ^{2 }⁄ 4
= 169 − 144
= 25.
Step4: Calculate Residual Sum of square using formula – Residual SS= Total SS− Treatment SS
here, Residual SS= 26 − 25 = 1
Step5 : Calculate degrees of freedom (df) for total, treatment, and residual. For total and treatment, it is n−1 and for residual, it is total degrees of freedom− treatment degrees of freedom.
here, treatment df= 2−1 = 1 and
residual df= (4−1) − (2−1)= 2
Step6: Calculate mean square for treatment and residual using formula Mean square = Sum of square/degrees of freedom.
here, Treatment mean square = 25/1 = 25
and Residual mean square = 1/2 = 0.5
Step7: Calculate variance ratio or F value using the formula – Mean square of treatment/ Mean square of Residual.
Step8: Draw ANOVA table including all the headlines below
Step9: Compare the calculated F value with that of the F distribution table and comment. If the calculated value is greater than the tabulated value, the null hypothesis is rejected. Thus there is a significant difference in the treatment. If it’s small, vise versa.
If the analysis is concerns with testing the difference between two sets of groups with further categorization according to two or more factors concerned we use two way ANOVA test. The steps for finding F value in 2 way ANOVA is quite different as in the one way as we have to calculate the treatment sum of square values for treatment and factors separately. So there will be more than one treatment value.
A simplified table is given as an example
Step1 and 2 are the same as in oneway ANOVA.
Step3: Calculate Treatment Sum of square for both treatments applied and factors using formula (total of 1st treatment)^{2}/n + …..+(total of last treatment)^{2}/n − (∑x) ^{2 }⁄ n.
here, Treatment sum of square for treatments (concentration) = (C1)^{2}/n + (C2)^{2}/n + (C3)^{2}/n − (∑x) ^{2 }⁄ n
and Treatment sum of square for factors = (F1)^{2}/n + (F2)^{2}/n + (F3)^{2}/n + (F4)^{2}/n − (∑x) ^{2 }⁄ n
Step4: Calculate Residual Sum of square using formula Residual SS = Total SS − (Treatment SS + Factor SS)
Step5: Calculate degrees of freedom for Total, Treatment, Factor, and Residual as same way done in one way ANOVA
Step6: Calculate Mean square for 2 Treatment Sum of squares for different treatments (concentration) and factor and for residual Sum of squares, using formula Mean square = Sum of square/degrees of freedom.
Step 7: Calculate variance ratio or F value for both Mean square values.
here, F value for treatments (concentration)= mean square of treatment (concentration)/ mean square of residual
and F value for factors= mean square of factors/ mean square of residual
Step8: Draw ANOVA table including all the headlines below
Step9: Compare the calculated F value with that of the F distribution table and comment. If the calculated value is greater than the tabulated value, the null hypothesis is rejected. Thus there is a significant difference in the treatment. If it’s small, vise versa.
F distribution table is given below. We will record the intercept value of horizontal (degrees of freedom for treatment) and vertical (degrees of freedom for residual) as tabulated F value.
Reference
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]]>The range consists of a bunch of data that shows the distinction between the highest and lowest values among the data set. So as to seek out the range, it’s necessary to first order the data from lowest to highest. Then we have to subtract the smallest value from the largest value in the set.
Mean deviation is a statistical measure of the average deviation of values from the mean within a set of samples. It’s calculated by finding the average of the observations. Then the difference of each observation from the mean is decided. The deviations then has got to be averaged. This analysis is mostly used to calculate how sporadic observations are from the mean.
2 5 7 10 12 14
Find out the average of these values by adding them and then and dividing them by the amount of values. In our example, the average is 8.3 (2+5+7+10+12+14=50, which is divided by 6).
Find the difference between each value and the average. Using our example, the differences are: 2 – 8.3 = 6.3 5 – 8.3 = 3.3 7 – 8.3 = 1.3 10 – 8.3 = 1.7 12 – 8.3 = 3.7 14 – 8.3 = 5.7
Find out the average of the differences by adding them and dividing by the number of observations. The average of the differences in this example is 3.66: (6.3+3.3+1.3+1.7+3.7+5.7 divided by 6).Get Free Netflix Now
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Variance
Variance is actually the expectation of the squared deviation of a variate from its mean. Informally, variance measures how far a group of numbers are spread out from their average value.
The variance is actually the square of the standard deviation.
*Variance value is usually positive[0 to (+ve)]
In statistics, the standard deviation is a calculation of the amount of variation or dispersion of a group of values. A low standard deviation directs that the values tend to be close to the mean of the set, on the other hand a high standard deviation indicates that the values are spread out over a wider range.
Whatever be the sampling procedure or the care taken while selecting the sample, the sample estimates of statistics, (X —, s or p) will differ from population parameters (μ, σ or P) because of chance or biological variability. Such a difference between sample and population values is measured by statistic know as sampling error or standard error or (SE).
**Standard error is thus a measure of chance variation and it doesn’t mean error or mistake.
The standard error is the measure of the approximate standard deviation of a statistical sample . The standard error measures the accuracy with which a sample distribution represents a population by using standard deviation. In statistics, a sample mean deviates from the actual mean of a population and this deviation is named the standard error of the mean.
The term “standard error” refers to the standard deviation of various sample statistics, such as the mean or median. For instance, the term “standard error of the mean” refers to the standard deviation of the distribution of sample means taken from a population. The smaller the standard error, the more representative the sample are going to be.
The relationship between the standard error and the standard deviation therefore is , for a given sample size, the standard error equals the standard deviation divided by the square root of the sample size. The standard error is reciprocally proportional to the sample size. The greater the sample size, the smaller the standard error because the statistic will approach the actual value.
The coefficient of variation is a statistical measure which shows the dispersion of data points within a data series around the mean. The coefficient of variation represents the ratio of the standard deviation to the mean and it is a useful statistic for comparing the degree of variation from one data series to a different one.
A confidence interval refers to the probability that a population parameter will fall between two set values for a certain proportion of times. Confidence intervals measure the degree of uncertainty or certainty in a sampling method and can take any number of probabilities with the foremost common being a 95% or 99% confidence level.
The empirical rule conjointly named as the threesigma rule or the 689599.7 rule provides a quick estimate of the spread of data in a normal distribution given the mean and standard deviation. Specifically the empirical rule indicates that for a normal distribution:
Q&A
1. What does the standard deviation measure?
2. According to the 689599.7 rule, or the empirical rule, if a data set has a normal distribution, approximately what percentage of data will be within one standard deviation of the mean?
3. A realtor tells you that the average cost of houses in a town is $176,000. You want to know how much the prices of the houses may vary from this average. What measurement do you need?
(A) standard deviation
(B) interquartile range
(C) variance
(D) percentile
(E) Choice (A) or (C)
4. What measure(s) of variation is/are sensitive to outliers?
(A) margin of error
(B) interquartile range
(C) standard deviation
(D) Choices (A) and (B)
(E) Choices (A) and (C)
5. You take a random sample of ten car owners and ask them, “To the nearest year, how old is your current car?” Their responses are as follows: 0 years, 1 year, 2 years, 4 years, 8 years, 3 years, 10 years, 17 years, 2 years, 7 years. To the nearest year, what is the standard deviation of this sample?
6. A sample is taken of the ages in years of 12 people who attend a movie. The results are as follows: 12 years, 10 years, 16 years, 22 years, 24 years, 18 years, 30 years, 32 years, 19 years, 20 years, 35 years, 26 years. To the nearest year, what is the standard deviation for this sample?
7. A large math class takes a midterm exam worth a total of 100 points. Following is a random sample of 20 students’ scores from the class:
Score of 98 points: 2 students
Score of 95 points: 1 student
Score of 92 points: 3 students
Score of 88 points: 4 students
Score of 87 points: 2 students
Score of 85 points: 2 students
Score of 81 points: 1 student
Score of 78 points: 2 students
Score of 73 points: 1 student
Score of 72 points: 1 student
Score of 65 points: 1 student
To the nearest tenth of a point, what is the standard deviation of the exam scores for the students in this sample?
8. A manufacturer of jet engines measures a turbine part to the nearest 0.001 centimeters. A sample of parts has the following data set: 5.001, 5.002, 5.005, 5.000, 5.010, 5.009, 5.003, 5.002, 5.001, 5.000. What is the standard deviation for this sample?
9. Two companies pay their employees the same average salary of $42,000 per year. The salary data in Ace Corp. has a standard deviation of $10,000, whereas Magna Company salary data has a standard deviation of $30,000. What, if anything, does this mean?
10. In which of the following situations would having a small standard deviation be most important?
(A) determining the variation in the wealth of retired people
(B) measuring the variation in circuitry components when manufacturing computer chips
(C) comparing the population of cities in different areas of the country
(D) comparing the amount of time it takes to complete education courses on the Internet
(E) measuring the variation in the production of different varieties of apple trees
11. Everyone at a company is given a yearend bonus of $2,000. How will this affect the standard deviation of the annual salaries in the company that year?
12. Calculate the sample variance and the standard deviation for the following measurements of weights of apples: 7 oz, 6 oz, 5 oz, 6 oz, 9 oz. Express your answers in the proper units of measurement and round to the nearest tenth.
13. Calculate the sample variance and the standard deviation for the following measurements of assembly time required to build an MP3 player: 15 min, 16 min, 18 min, 10 min, 9 min. Express your answers in the proper units of measurement and round to the nearest whole number.
14. Which of the following data sets has the same standard deviation as the data set with the numbers 1, 2, 3, 4, 5? (Do this problem without any calculations!)
(A) Data Set 1: 6, 7, 8, 9, 10
(B) Data Set 2: –2, –1, 0, 1, 2
(C) Data Set 3: 0.1, 0.2, 0.3, 0.4, 0.5
(D) Choices (A) and (B)
(E) None of the data sets gives the same standard deviation as the data set 1, 2, 3, 4, 5.
1. (Ans) how concentrated the data is around the mean
A standard deviation measures the amount of variability among the numbers in a data set. It calculates the typical distance of a data point from the mean of the data. If the standard deviation is relatively large, it means the data is quite spread out away from the mean. If the standard deviation is relatively small, it means the data is concentrated near the mean.
2. ( Ans) approximately 68%
According to the empirical rule, the bellshaped curve of a normal distribution will have 68% of the data points within one standard deviation of the mean.
3. (Ans) E. Choice (A) or (C) (standard deviation or variance)
The standard deviation is a way of measuring the typical distance that data is from the mean and is in the same units as the original data. The variance is a way of measuring the typical squared distance from the mean and isn’t in the same units as the original data. Both the standard deviation and variance measure variation in the data, but the standard deviation is easier to interpret.
4. (Ans) E. Choices (A) and (C) (margin of error; standard deviation)
The standard deviation measures the typical distance from the data to the mean (using all the data to calculate). Outliers are far from the mean, so the more outliers there are, the higher the standard deviation will be. You calculate the margin of error by using the sample standard deviation so it’s also sensitive to outliers. The interquartile range is the range of the middle 50% of the data, so outliers won’t be included, making it less sensitive to outliers than the standard deviation or margin of error
5. (Ans) 5 years
The formula for the sample standard deviation of a data set is
s=√[∑(x− ̄x)²/ n−1 ]where x is a single value, ̄x is the mean of all the values, represents the sum of the squared differences from the mean, and n is the sample size.
First, find the mean of the data set by adding together the data points and then dividing by the sample size (in this case, n = 10):
̄x =(0+1+2+4+8+3+10+17+2+7) /10
=54 /10
=5.4
Then, subtract the mean from each number in the data set and square the differences, (x – ̄x)²: (0 – 5.4)² = (–5.4)²= 29.16
(1 – 5.4)²= (–4.4)² = 19.36
(2 – 5.4)² = (–3.4)² = 11.56
(4 – 5.4)² = (–1.4)² = 1.96
(8 – 5.4)² = (2.6)² = 6.76
(3 – 5.4)² = (–2.4)² = 5.76
(10 – 5.4)²= (4.6)²= 21.16
(17 – 5.4)² = (11.6)²= 134.56
(2 – 5.4)² = (–3.4)²= 11.56
(7 – 5.4)²= (1.6)² = 2.56
Next, add up the results from the squared differences: 29.16 + 19.36 + 11.56 + 1.96 + 6.76 + 5.76 + 21.16 + 134.56 + 11.56 + 2.56 = 244.4 Finally, plug the numbers into the formula for the sample standard deviation:
s=
√{∑(x− ̄x)2 /n−1} =√(244.4/ 10−1) =√27.156 =5.21 The question asks for the nearest year, so round to 5 years.
6. (Ans) 8 years
The formula for the sample standard deviation of a data set is
s=√{∑(x− ̄x)²/ n−1} where x is a single value, ̄x is the mean of all the values, represents the sum of the squared differences from the mean, and n is the sample size
First, find the mean of the data set by adding together the data points and then
dividing by the sample size (in this case, n = 12):
̄x =(12+10+16+22+24+18+30+32+19+20+35+26)/ 12
=264 /12
=22
Then, subtract the mean from each number in the data set and square the differences, (x – ̄x)2:
(12 – 22)² = (–10)² = 100
(10 – 22)²= (–12)² = 144
(16 – 22)² = (–6)² = 36
(22 – 22)² = (0)² = 0
(24 – 22)² = (2)² = 4
(18 – 22)²= (4)² = 16
(30 – 22)² = (8)²= 64
(32 – 22)² = (10)² = 100
(19 – 22)²= (–3)² = 9
(20 – 22)² = (–2)² = 4
(35 – 22)² = (13)² = 169
(26 – 22)² = (4)² = 16
Next, add up the results from the squared differences: 100 + 144 + 36 + 0 + 4 + 16 + 64 + 100 + 9 + 4 + 169 + 16 = 662 Finally, plug the numbers into the formula for the sample standard deviation:
s=√{∑(x− ̄x)²/ n−1} =√(662 /12−1) =√60.1818 =7.76 The question asks for the nearest year, so round to 8 years.
7. (Ans) 8.7 points
The formula for the sample standard deviation of a data set is
s=√[∑(x− ̄x)²/ n−1 ]where x is a single value, ̄x is the mean of all the values, represents the sum of the squared differences from the mean, and n is the sample size.
First, find the mean of the data set. Although you don’t have a list of all the individual values, you do know the test score for each student in the sample. For example, you know that three students scored 92 points, so if you listed every student’s score individually, you’d see 92 three times, or (92)(3). To find the mean this way, multiply each exam score by the number of students who received that score, add the products together, and then divide by the number of students in the sample (n = 20):
(98)(2) = 196
(95)(1) = 95
(92)(3) = 276
(88)(4) = 352
(87)(2) = 174
(85)(2) = 170
(81)(1) = 81
(78)(2) = 156
(73)(1) = 73
(72)(1) = 72
(65)(1) = 65
̄x =196+95+276+352+174+170+81+156+73+72+65 /20
=1,710/ 20
=85.5
Next, subtract the mean from each different exam score in the data set and square the differences, (x – ̄x)². Note: There are 11 different exam scores here — 98, 95, 92, 88, 87, 85, 81, 78, 73, 72, and 65 — but 20 students. First, work with the 11 exam scores.
(98 – 85.5)2 = (12.5)2 = 156.25
(95 – 85.5)2 = (9.5)2 = 90.25
(92 – 85.5)2 = (6.5)2 = 42.25
(88 – 85.5)2 = (2.5)2 = 6.25
(87 – 85.5)2 = (1.5)2 = 2.25
(85 – 85.5)2 = (–0.5)2 = 0.25
(81 – 85.5)2 = (–4.5)2 = 20.25
(78 – 85.5)2 = (–7.5)2 = 56.25
(73 – 85.5)2 = (–12.5)2 = 156.25
(72 – 85.5)2 = (–13.5)2 = 182.25
(65 – 85.5)2 = (–20.5)2 = 420.25
Now, multiply each value by the number of students who got that score: (156.25)(2) = 312.5
(90.25)(1) = 90.25
(42.25)(3) = 126.75
(6.25)(4) = 25
(2.25)(2) = 4.5
(0.25)(2) = 0.5
(20.25)(1) = 20.25
(56.25)(2) = 112.5
(156.25)(1) = 156.25
(182.25)(1) = 182.25
(420.25)(1) = 420.25
Then, add up those results: 312.5 + 90.25 + 126.75 + 25 + 4.5 + 0.5 + 20.25 + 112.5 + 156.25 + 182.25 + 420.25 = 1,451 Finally, plug the numbers into the formula for the sample standard deviation
s=√{(xx¯)/n1}=√(1451/201 )=√76.37=8.74
The question asks for the nearest tenth of a point, so round to 8.7.
8. (Ans) 0.0036 cm
The formula for the sample standard deviation of a data set is
s=√[∑(x− ̄x)²/ n−1 ]
where x is a single value, ̄x is the mean of all the values, represents the sum of the squared differences from the mean, and n is the sample size.
First, find the mean of the data set by adding together the data points and then dividing by the sample size (in this case, n = 10):
̄x =(5.001+5.002+5.005+5.010+5.009+5.003+5.002+5.001+5.000)/ 10
=50.033/ 10
=5.0033
Then, subtract the mean from each number in the data set and square the differences, (x – ̄x)²:
(5.001 – 5.0033)² = (–0.0023)² = 0.00000529
(5.002 – 5.0033)² = (–0.0013)² = 0.00000169
(5.005 – 5.0033)² = (0.0017)² = 0.00000289
(5.000 – 5.0033)² = (–0.0033)² = 0.00001089
(5.010 – 5.0033)² = (0.0067)² = 0.00004489
(5.009 – 5.0033)² = (0.0057)² = 0.00003249
(5.003 – 5.0033)² = (–0.0003)² = 0.00000009
(5.002 – 5.0033)² = (–0.0013)² = 0.00000169
(5.001 – 5.0033)² = (–0.0023)² = 0.00000529
(5.000 – 5.0033)² = (–0.0033)² = 0.00001089
Next, add up the results from the squared differences: 0.00000529 + 0.00000169 + 0.00000289 + 0.00001089 + 0.00004489 + 0.00003249 + 0.00000009 + 0.00000169 + 0.00000529 + 0.00001089 = 0.0001161 Finally, plug the numbers into the formula for the sample standard deviation:
s=√{∑(x− ̄x)²/n−1} =√(0.0001161 /10−1 )=√0.0000129 =0.0036 The sample standard deviation for the jet engine turbine part is 0.0036 centimeters.
9. (Ans) There is more variation in salaries in Magna Company than in Ace Corp.
The larger standard deviation in Magna Company shows a greater variation of salaries in both directions from the mean than Ace Corp. The standard deviation measures on average how spread out the data is (for example, the high and low salaries at each company).
10. (Ans) B. measuring the variation in circuitry components when manufacturing computer chips
The quality of the vast majority of manufacturing processes depends on reducing variation to as little as possible. If a manufacturing process has a large standard deviation, it indicates a lack of predictability in the quality and usefulness of the end product.
11. (Ans) There will be no change in the standard deviation.
All the data points will shift up $2,000, and as a result, the mean will also increase by $2,000. But each individual salary’s distance (or deviation) from the mean will be the same, so the standard deviation will stay the same.
12. (Ans) The sample variance is 2.3 ounces². The standard deviation is 1.5 ounces.
You find the sample variance with the following formula: s² = ∑(x− ̄x)²/n−1 where x is a single value, ̄x is the mean of all the values, represents the sum of the squared difference scores, and n is the sample size. First, find the mean by adding together the data points and dividing by the sample size (in this case, n = 5):
̄x =7+6+5+6+9 5
=33 5
=6.6 Then, subtract the mean from each data point and square the differences, (x− ̄x)²:
(7 – 6.6)² = (0.4)² = 0.16
(6 – 6.6)² = (–0.6)² = 0.36
(5 – 6.6)²= (–1.6)² = 2.56
(6 – 6.6)² = (–0.6)² = 0.36
(9 – 6.6)² = (2.4)² = 5.76
Next, plug the numbers into the formula for the sample variance:
The sample variance is 2.3 ounces². But these units don’t make sense because there’s no such thing as “square ounces.” However, the standard deviation is the square root of the variance, so it can then be expressed in the original units: s = 1.5 ounces (rounded). For this reason, standard deviation is preferred over the variance when it comes to measuring and interpreting variability in a data set.
13. (Ans) The sample variance is 15 minutes². The standard deviation is 4 minutes.
The formula for the sample standard deviation of a data set is
s=√[∑(x− ̄x)²/ n−1 ]
where x is a single value, ̄x is the mean of all the values, represents the sum of the squared differences from the mean, and n is the sample size.
First, find the mean by adding together the data points and dividing by the sample size (in this case, n = 5):
̄x =15+16+18+10+9/ 5
=68/ 5
=13.6 Then, subtract the mean from each data point and square the differences, (x− ̄x)2:
(15 – 13.6)2 = (1.4)2 = 1.96
(16 – 13.6)2 = (2.4)2 = 5.76
(18 – 13.6)2 = (4.4)2 = 19.36
(10 – 13.6)2 = (–3.6)2 = 12.96
(9 – 13.6)2 = (–4.6)2 = 21.16
Next, plug the numbers into the formula for the sample variance:
s² = ∑(x− ̄x)²/n−1
=1.96+5.76+19.36+12.96+21.16 /5−1
=61.2/ 4
=15.3
The sample variance is 15.3 minutes². But these units don’t make sense because there’s no such thing as “square minutes.” However, the standard deviation is the square root of the variance, so it can then be expressed in the original units: s = 3.91 minutes (rounded up to 4). For this reason, standard deviation is preferred over the variance when it comes to measuring and interpreting variability in a data set.
14. (Ans) D. Choices (A) and (B) (Data Set 1; Data Set 2)
The original data set contains the numbers 1, 2, 3, 4, 5. Data Set 1 just shifts those numbers up by five units to get 6, 7, 8, 9, 10. Standard deviation represents typical (or average) distance from the mean, and although the mean in Data Set 1 changes from 3 to 8, the distances from each point to that new mean stay the same as they were for the original data set, so the average distance from the mean is the same. Data Set 2 contains the numbers –2, –1, 0, 1, 2. These numbers shift the original data set’s values down by three units. For example, 1 – 3 = –2, 2 – 3 = –1, and so forth. Therefore, the standard deviation doesn’t change from the original data set. Data Set 3 divides all the numbers in the original data set by 10, making them closer to the mean, on average, than the original data set. Therefore, the standard deviation is smaller.
Reference
Statistics: 1,001 Practice Problems For Dummies.
Revised by
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]]>The post Organisation and Presentation of Data appeared first on Plantlet.
]]>There are two main methods of presenting data of a variable character or a variable.
A) Tabulation/Tabular Presentation
B) Drawing/Graphical Presentation
Tabulation is a device for presenting data from a mass of statistical data. Preparation of frequency distribution table is the first requirement for that. Tables are often simple or complex depending upon the measurement of a single set of things or multiple sets of things.
Frequency Distribution
The distribution of the overall number of observations among the various categories is termed as frequency distribution.
Frequency Distribution is a very important step in statistical analysis. It groups a sizable amount of series or observations of the master table and presents the data very concisely, giving all information at a look.
It records how frequently a characteristic or an occurrence occurs in persons of the same group. Data are often recorded within the sort of frequency table.
In short, collecting and summarizing a great amount of data is called frequency distribution.
Rules for Making Frequency Distribution Table
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1. Simple Tabulation
Simple Tabulation is when the information/data are tabulated to one characteristic. For example, the survey determined the frequency or number of employees of a firm owning different brands of mobile phones.
2. Complex tabulation
When the data are tabulated consistently with many characteristics, it is said to be a complex tabulation.
According to the type of data, graphical distribution or drawing is categorized as follows:
1) Histogram
A histogram is a graphical display of data using bars of various heights. In a histogram, each bar groups numbers into ranges. Taller bars show that more data falls in this range. A histogram displays the form/shape and spread of continuous sample data.
2) Frequency Polygon
A frequency polygon is a graph constructed by using lines to join the midpoints of every interval or bin. The heights of the points depict the frequencies. A frequency polygon is usually created from the histogram or by calculating the midpoints of the bins from the frequency distribution table.
3) Frequency Curve
A frequency curve is a smooth curve for which the entire area is taken to be unity. It’s a limiting sort of a histogram or frequency polygon. The frequency curve for distribution is obtained by drawing a smooth and freehand/blank check curve through the midpoints of the upper sides of the rectangles forming the histogram.
4) Line Chart
A line chart is a graphical representation of an asset’s historical price action that connects a series of data points with a continual line. This is often the foremost basic type of chart used in finance and typically only depicts a security’s closing prices over time.
5) Normal Distribution Curve
A normal distribution is a type of continuous probability distribution for a realvalued random variable. A normal distribution is usually informally called a bell curve.
6) Cumulative Distribution Curve
In statistics, the cumulative distribution function of a realvalued random variable, or just distribution function of, evaluated at, is the probability that will take a value less than or adequate to.
7) Scatter Diagram
A graph during which the values of two variables are plotted along two axis, the pattern of the resulting points revealing any correlation present.
1) Bar Chart
A bar chart or bar graph is a chart or graph that presents categorical data with rectangular bars with heights or lengths proportional to the values that they represent. The bars can be often plotted vertically or horizontally. A vertical bar chart is usually called a column chart.
2) Pictogram
A pictogram is a chart that uses pictures to represent data. Pictograms are set out in the same way as to bar charts, but rather than bars they use columns of pictures to point out the numbers involved.
3) Pie Chart
A pie chart is a sort of graph in which a circle is split into sectors that each represents a proportion of the entire. Pie charts are a useful way to organize data in order to see the size of components relative to the entire and are particularly good at showing percentage or proportional data.
4) Map Diagram
A map diagram is a way of representation of any event distribution by means of diagrams, that are placed on the map inside the structure of territorial division which expresses the summarized value of this event within the bounds of this territorial structure.
You Are interested in the percentage of female versus male shoppers at a department store.
Which chart or graph would be appropriate to display the proportion of males versus females among the shoppers?
A)A bar graph
B)A time plot
C)A pie chart
D)Choices (A) and (C)
E)Choices (A),(B) and (C)
Ans: (D)
Gender is a qualitative variable, so both bar graphs and pie charts are appropriate to display the proportion of males versus females among the shoppers. You could use a time plot only if you knew how many males and how many females were in the store at each individual time period.
Ref:Statistics: 1,001 Practice Problems For Dummies
Source:
1. Feature image: https://epthinktank.eu/2015/08/17/eprsgraphicwarehousewhenapicturetells/amp/
2. Methods of Biostatistics Mahajan
Revised by:
1. Tarek siddiki Taki ( 08.08.2020)
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]]>There are various methods to measure the central value of an observation. For instance, average, mean, median, mode, etc.
The average value of a characteristic is the one central value around which all other observations are distributed. In any large series, nearly 50% of observations lie above the central value whereas the other 50% lie below the central value. It indicates how the values lie close to the center.
**Average is the measure that indicates the central tendency or concentration of all other observations around the central value.
There are three types of averages or measures of central position or central tendency mean, median, and mode.
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This is an arithmetic average of a collection of values. It is the sum of observations divided by the number of observations.
Let,the observations are x_{1,}x_{2,}x_{3,}….
Mean=∑x/n (n=number of observations)
Mean in case of continuous series:
Number of organisms  frequency(f)  midpoint(m)  f×m 
59  3  7  21 
1014  5  12  60 
1519  6  17  102 
2024  4  22  88 
2529  3  27  81 
∑f=21  ∑fm=352 
x¯=∑fm/∑f=352/21=16.76
When all the observations of a variable one arranged in either ascending or descending order, the middle observation is known as the median.
The median is a locational average & it is that value that divides the series into two equal shares in a series of observations. The median equally divides the number of items of observations.
The Median will be the average of two values according to the following formula.
when,n=10
n/2=10/2=5th
(n+2)/2=(10+2)/2=6th, Here median will be the average of 5th & 6th value of the series.
Example:4,6,7,8,8,9,11,12,12,14 Here,5th value=8 & 6th value=9
Now,median=(8+9)/2=17/2=8.5
Number of nodules/root:7,6,9,8,9,10,6,7,10
Arranged in ascending order:6,6,7,7,8,9,9,10,10
Median=(n+1)/2=(9+1)/2=5th value=8
Time (Sec) Class Interval 
Frequency  Cumulative Frequency 
3035  3  3 
3641  10  13 
4247  18  31 
4853  25  56 
5459  8  64 
6065  6  70 
∑n=70 
Median=L+{(n/2Fc)/fm}×h=48+{(70/231)/25}×6=48.96
L= lower value of the highly frequent class (4853)
n=number of observations [n=70],[2f=70]
Fc=cumulative value of the previous class of the median class [Fc=31]
fm= frequency of median class
h=class interval[h=6]
This is the most frequently occurring observation in a series.
Mode is the type of average that refers to the most common or most regularly occurring value in a series of data.
Class  Frequency 
3140  4 
4150  6 
5160  8 
6170  12 
7180  9 
8190  7 
91100  4 
Mode=L+{f1/(f1+f2)}×h=61+{4/(4+3)}×10=61+0.57×9=66.71
Here,
L=lower value of the modal class=61
f1=frequency of the modal class – frequency of the previous class=128=4
f2=frequency of the modal class – frequency of the next class=129=3
h=class interval=10
**Out of three measures of central tendency mean is better and utilized more often because it uses all the observations in the data and is further used in the tests of significance
Symmetrical distribution is a distribution in which the values of variables occur at regular frequencies, and the mean, median, and mode occur at the same point. Unlike asymmetrical distribution, symmetrical distribution does not skew.
**In short, the distributions in which both sides are the same(mirror image) about the middle ordinate is called symmetrical distribution.
Asymmetrical distribution is a distribution in which the values of variables occur at irregular frequencies and the mean, median, and mode occur at different points. An asymmetric distribution exhibits skewness.
Some distributions are steep on one side and have a long tail on the other side. This characteristic of a distribution is called skewness.
If the tail is on the right side the distribution is said to be skewed right. This type of skewness is called positive skewness.
**Biological phenomena are very commonly skewed right.
If the tail is on the left side, the distribution is said to be skewed left. This type of skewness is called negative skewness.
**Skewness is a measure of symmetry, or to say more certainly, the lack of symmetry.
Similar to skewness, kurtosis is a statistical measure that is used to describe the distribution.
Kurtosis is a measure of whether or not the data are heavytailed or lighttailed relative to a normal distribution. That is, data sets with high kurtosis tend to own heavy tails or outliers. Data sets with low kurtosis tend to own light tails or lack of outliers.
**Kurtosis=3 depicts the normal distribution
In statistics, an outlier is a data point that differs considerably from other observations. An outlier may be due to variability in the measurement or it may suggest experimental error; the latter are generally excluded from the data set. An outlier can cause serious issues in statistical analyses.
1. Which of the following data sets has a median of 3?
(A) 3, 3, 3, 3, 3
(B) 2, 5, 3, 1, 1
(C) 1, 2, 3, 4, 5
(D) 1, 2, 4, 4, 4
(E) Choices (A) and (C)
2. To the nearest tenth, what is the mean of the following data set? 14, 14, 15, 16, 28, 28, 32, 35, 37, 38
3. To the nearest tenth, what is the mean of the following data set? 15, 25, 35, 45, 50, 60, 70, 72, 100
4. To the nearest tenth, what is the mean of the following data set? 0.8, 1.8, 2.3, 4.5, 4.8, 16.1, 22.3
5. To the nearest thousandth, what is the mean of the following data set? 0.003, 0.045, 0.58, 0.687, 1.25, 10.38, 11.252, 12.001
6. To the nearest tenth, what is the median of the following data set? 6, 12, 22, 18, 16, 4, 20, 5, 15
7. To the nearest tenth, what is the median of the following data set? 18, 21, 17, 18, 16, 15.5, 12, 17, 10, 21, 17
8. To the nearest tenth, what is the median of the following data set? 14, 2, 21, 7, 30, 10, 1, 15, 6, 8
9. To the nearest hundredth, what is the median of the following data set? 25.2, 0.25, 8.2, 1.22, 0.001, 0.1, 6.85, 13.2
10. Compare the mean and median of a data set that has a distribution that is skewed right.
11. Compare the mean and median of a data set that has a distribution that is skewed left.
12. Compare the mean and the median of a data set that has a symmetrical distribution.
13. Which measure of center is most resistant to (or least affected by) outliers?
1. Ans) E. Choices (A) and (C) (3, 3, 3, 3, 3; 1, 2, 3, 4, 5)
To find the median, put the data in order from lowest to highest, and find the value in the middle. It doesn’t matter how many times a number is repeated. In this case, the data sets 3, 3, 3, 3, 3 and 1, 2, 3, 4, 5 each have a median of 3.
2. Ans) 25.7
Use the formula,
¯x=∑x/n
where ̄x is the mean, ∑ represents the sum of the data values, and n is the number of values in the data set. In this case, x = 14 + 14 + 15 + 16 + 28 + 28 + 32 + 35 + 37 + 38 = 257, and n = 10. So the mean is 257 10 =25.
3. Ans) 52.4
Use the formula for calculating the mean
¯x=∑x/n
where ̄x is the mean, ∑ represents the sum of the data values, and n is the number of values in the data set. In this case, x = 15 + 25 + 35 + 45 + 50 + 60 + 70 + 72 + 100 = 472, and n = 9. So the mean is 472 9 =52.4444
The question asks for the nearest tenth, so you round to 52.4.
4. Ans) 7.5
Use the formula for calculating the mean
¯x=∑x/n
where ̄x is the mean, ∑ represents the sum of the data values, and n is the number of values in the data set. In this case, x = 0.8 + 1.8 + 2.3 + 4.5 + 4.8 + 16.1 + 22.3 = 52.6, and n = 7. So the mean is 52.6 7 =7.5143
The question asks for the nearest tenth, so you round to 7.5.
5. Ans) 4.525
Use the formula for calculating the mean
¯x=∑x/n
where ̄x is the mean, ∑ represents the sum of the data values, and n is the number of values in the data set. In this case, x = 0.003 + 0.045 + 0.58 + 0.687 + 1.25 + 10.38 + 11.252 + 12.001 = 36.198, and n = 8. So the mean is 36.198 8 =4.52475
The question asks for the nearest thousandth, so you round to 4.525.
6. Ans) 15.0
To find the median, put the numbers in order from smallest to largest: 4, 5, 6, 12, 15, 16, 18, 20, 22 Because this data set has an odd number of values (nine), the median is simply the middle number in the data set: 15.
7. Ans) 17.0
To find the median, put the numbers in order from smallest to largest: 10, 12, 15.5, 16, 17, 17, 17, 18, 18, 21, 21 Because this data set has an odd number of values (11), the median is simply the middle number in the data set: 17.
8. Ans) 9.0
To find the median, put the numbers in order from smallest to largest: 1, 2, 6, 7, 8, 10, 14, 15, 21, 30 Because this data set has an even number of values (ten), the median is the average of the two middle numbers: 8+10/2 =9.0
9. Ans) 4.04
To find the median, put the numbers in order from smallest to largest. 0.001, 0.1, 0.25, 1.22, 6.85, 8.2, 13.2, 25.2 Because this data set has an even number of values (eight), the median is the average of the two middle numbers: 1.22+6.85/2 =4.035 The question asks for the nearest hundredth, so round to 4.04.
10. Ans) The mean will have a higher value than the median.
A data set distribution that is skewed right is asymmetrical and has a large number of values at the lower end and few numbers at the high end. In this case, the median, which is the middle number when you sort the data from smallest to largest, lies in the lower range of values (where most of the numbers are). However, because the mean finds the average of all the high and low values, the few outlying data points on the high end cause the mean to increase, making it higher than the median.
11. Ans) The mean will have a lower value than the median.
A data set distribution that is skewed left is asymmetrical and has a large number of values at the high end and few numbers at the low end. In this case, the median, which is the middle number when you sort the data from smallest to largest, lies in the upper range of values (where most of the numbers are). However, because the mean finds the average of all the high and low values, the few outlying data points on the low end cause the mean to decrease, making it lower than the median.
12. Ans) The mean and median will be fairly close together.
When a data set has a symmetrical distribution, the mean and the median are close together because the middle value in the data set, when ordered smallest to largest, resembles the balancing point in the data, which occurs at the average.
13. Ans) median
The median is the middle value of the data points when ordered from smallest to largest. When the data is ordered, it no longer takes into account the values of any of the other data points. This makes it resistant to being influenced by outliers. (In other words, outliers don’t really affect the median.) In contrast, the mean takes every specific data value into account. If the data points contain some outliers that are extreme values to one side, the mean will be pulled toward those outliers.
Reference
Statistics: 1,001 Practice Problems For Dummies
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]]>
Objectives of this articleTo understand

Read the next part of this article: Introduction to Biostatistics part 2
Statistics is defined as a body of processes for creating reasonable and wise decisions in the face of uncertainty. These are applied in the analysis of numerical data of various aspects including interpretation of data based on certain statistical principles.
Statistics is a field of study involving techniques or methods of collecting data, classification, summarizing, interpretation, drawing, inferences, testing of hypotheses, making recommendations, etc. only when a part of data is used.
Biostatistics is the term referred to when tools of statistics are applied to the data that is derived from biological sciences. In other words, when the principles of statistics are applied to the study of organisms or living systems, the study is called biostatistics or biometry.
It encompasses the design of biological experiments, especially in medicine, pharmacy, agriculture, and fishery; the collection, summarization, and analysis of data from those experiments; and the interpretation and inference from the obtained results.
Reference
In statistical language, any character, characteristic, or quality that varies is called a variable.
A characteristic that takes on different values in different persons, places, or things such as height, weight, blood pressure, age, etc is variable. It is denoted usually as “x”. Variables can be of two types: Categorical & Numerical variables.
A random variable is a variable whose value is a numerical outcome of a random phenomenon. For example: Flip three coins and let x represent the number of heads. Here, x is a random variable.
A random variable is not a probability. Its value doesn’t need to be positive or between 0 and 1 as in the case of probability.
Numerical variables are divided into two categories:
Qualitative Data are classified by counting the individuals or things having the same characteristic or attribute; and not by measurement. Examples:
Individuals with the same characteristics are counted to form specific groups or classes.
Qualitative data are discrete, such as the number of deaths in different years, the population of different towns, and persons with different blood groups in a population.
A continuous variable is a variable that has an infinite number of possible values. In other words, any value is possible for the variable.
A continuous variable doesn’t have to have every possible number (like infinity to +infinity), it can also be continuous between two numbers, like 1 and 2. For example, data of a discrete variable could be 1, 2 while the continuous variables could be 1, 2 and also everything or anything in between: 1.00, 1.01, 1.001, 1.0001…
Examples
The weight of students from 2nd year are (in kg) 40.9, 45, 55, 50.1, 53, 54, 54, 48, 48.5, 46, 70, 85, 82, 83.1, 62.5 etc.(See how the number varies within a range)
In the case of quantitative/continuous data, there are two variables the characteristics such as height & frequency. We find the characteristic, as well as the frequency both, vary from person to person as well as from group to group.
The quantitative data obtained from characteristic variables (e.g. height of individuals in 2nd year) are called continuous data because each individual has one measurement from a continuous spectrum or range.
Some of the statistical methods employed in the analysis of quantitative data are mean, range, standard deviation, coefficient of variation, etc.
References
Revised by
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]]>
Objectives of this articleTo understand

Read the previous article: Introduction to biostatistics part 1
In statistics, a population is a set of homogeneous items or events that is of interest for some question or experiment.
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Example: Birth weights of all babies in a particular hospital in month May, the monthly expenditure of nonresidential students of 2nd year in Department of Botany, etc.
A sample is a smaller group of members of a population selected as a representative of the population.
In statistical inference, a subset or a portion of the population (a statistical sample) is chosen to represent the population in statistical analysis. If a sample is chosen properly, characteristics of the complete population that the sample is drawn from can be estimated from corresponding characteristics of the sample.
Selecting samples being unbiased is called random sampling. A very important feature of a good study is that the sample is randomly selected from the target population. Randomly means that every member of the target population has an identical chance of being included in the sample. In other words, the method you use for selecting your sample can not be biased.
It’s important because if you select your subjects in a biased way, then your results will also be biased. And more likely, it wouldn’t represent the population.
Read more about Random Sampling
Suppose you’re making a phone survey on the job satisfaction of Bangladeshis. If you call them at home during the day between 8 a.m. and 5 p.m. you’ll miss out on all those who work or has a job during the day; it could be that day workers are more satisfied than night workers.
Or your boss told you to make a survey of the overall development of the office from the year 2010 to 2020. To make your boss happy while sampling you chose the things as samples that actually developed but didn’t count those which almost remained the same or demoted in 10 years.
Like the abovementioned examples, there are many ways of being biased. To get a good survey it’s necessary to avoid all kinds of biasness possible.
References
Mahajan BK 2002. (Methods in Biostatistics)(6th edition)
Deborah Rumsey (Statistics for Dummies)
You want to calculate the percentage of female versus male shoppers at a department store. So on a Saturday morning, you place data collectors at each of the store’s four entrances for three hours, and you have them record how many men and women enter the store during that time.
1. Why collecting data at the store on one Saturday morning for three hours can cause bias in the data?
Ans: E. All of these choices are true.
Bias is systematic favoritism in the data. You want to get data that represents all customers at the store, irrespective of what day or what time they shop, whether they shop in couples or alone, and so on. You can’t assume that the people who shopped during those three hours on that Saturday morning are representative of the store’s total clients. This sample wasn’t drawn randomly — everyone who walked in was counted.
Parameter is a summary value that describes the population such as its mean, variance, correlation coefficient, proportion, etc.
Statistic is a summary value that describes the sample as its mean, standard deviation, standard error, correlation coefficient, proportion, etc.
The value is calculated from the sample and is often applied to the population but may or may not be a valid estimate of the population.
There is a good reason that the population parameter and sample static will vary; hopefully very little but can be significant too if the sample is biased or too small (i.e. doesn’t represent the population properly).
Q&A
You’re assigned to a job where you have to know what percent of all households in a large city has a single woman as the head of the household. To calculate this percentage, you survey 200 households and determine how many of these 200 are headed by a single woman.
1. In this example, what is the population?
2. In this example, what is the sample?
3. In this example, what is the parameter?
4. In this example, what is the statistic?
1. Ans: All households in the city.
A population is a complete or whole group you’re interested in studying. The aim here is to estimate what percent of all households in a large city has a single woman as the head of the household. The population is total households and the variable is whether a single woman runs the household.
2. Ans: Selected 200 households.
The sample is a portion or subset drawn from the entire population you’re interested in studying. In this inference, the sample is the 200 households selected out of all the households in the city.
3. Ans: The percent of households headed by single women in the city.
A parameter is some characteristic of the population. Because studying a population directly isn’t generally possible, parameters are usually estimated by using statistics (numbers calculated from sample data).
4. Ans: The percent of households headed by single women within the 200 selected households.
A statistic is a number representing some characteristic that you calculate from your sample data; the statistic is used to estimate the parameter (the same characteristic in the population).
Reference
Statistics: 1,001 Practice Problems For Dummies
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