Solved Problems on Population Variance
Problem 1: Suppose we have the heights (in centimeters) of five students: 160, 165, 170, 175, and 180. The mean height is 170 cm. Calculate the population variance.
Solution:
Calculate the squared deviations from the mean:
[Tex](160 β 170)^2 = 100\\ (165 β 170)^2 = 25\\ (170 β 170)^2 = 0\\ (175 β 170)^2 = 25\\ (180 β 170)^2 = 100[/Tex]
Sum up the squared deviations = 100 + 25 + 0 + 25 + 100 = 250
Divide by the total number of observations (which is 5) = 250 / 5 = 50
Therefore, the population variance for this data set is 50 square centimeters.
Problem 2: Suppose we have the following exam scores (out of 100) for a class of 10 students: 78, 85, 92, 70, 88, 95, 81, 79, 90, and 84. The mean score is 84. Calculate the population variance.
Solution:
Calculate the squared deviations from the mean:
[Tex](78 β 84)^2 = 36\\ (85 β 84)^2 = 1\\ (92 β 84)^2 = 64\\ (70 β 84)^2 = 196\\ (88 β 84)^2 = 16\\ (95 β 84)^2 = 121\\ (81 β 84)^2 = 9\\ (79 β 84)^2 = 25\\ (90 β 84)^2 = 36\\ (84 β 84)^2 = 0[/Tex]
Sum up the squared deviations: (36 + 1 + 64 + 196 + 16 + 121 + 9 + 25 + 36 + 0 = 504)
Divide by the total number of observations (which is 10): (504 / 10 = 50.4)
Therefore, the population variance for this data set is 50.4.
Problem 3: Consider daily maximum temperatures (in degrees Celsius) recorded over a week: 28, 30, 29, 31, 27, 28, and 32. The mean temperature is 29. Calculate the population variance.
Solution:
Calculate the squared deviations from the mean:
[Tex](28 β 29)^2 = 1\\ (30 β 29)^2 = 1\\ (29 β 29)^2 = 0\\ (31 β 29)^2 = 4\\ (27 β 29)^2 = 4\\ (28 β 29)^2 = 1\\ (32 β 29)^2 = 9[/Tex]
Sum up the squared deviations = 1 + 1 + 0 + 4 + 4 + 1 + 9 = 20
Divide by the total number of observations (which is 7) = [Tex]20 / 7 \approx 2.86[/Tex]
Therefore, the population variance for this temperature data set is approximately 2.86.
Population Variance
Population variance is a fundamental concept in statistics that quantifies the average squared deviation from the mean of a set of data points in a population. It is a measure of how spread out a group of data points is.
There are two types of data available, namely, ungrouped and grouped data. Thus, there are two formulas to calculate the population variance. In this article, we will learn more about population variance, its formulas, and various associated examples.
Table of Content
- What is Population Variance?
- Formula of Population Variance
- Ungrouped Data
- Grouped Data
- Population Variance and Sample Variance
- Population Variance and Standard Deviation
- Solved Problems on Population Variance
- Practice Questions on Population Variance
- FAQs on Population Variance