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VOOZH | about |
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VOOZH | about |
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
Population variance determines how far each data point is from the population mean. It can be defined as the average of the square of the deviations from the data’s mean value. If all data points are very close to the mean, the variance will be small; if data points are spread out over a wide range, the variance will be larger.
The population variance is a fundamental statistical measure that quantifies the dispersion or variability of a dataset around its mean. Whether dealing with grouped or ungrouped data, understanding the population variance formula is essential for analyzing and interpreting the spread of data points within a population.
Ungrouped data, also known as raw data, consists of individual data points that are not categorized or grouped into intervals. Each data point in ungrouped data represents a distinct value or observation.
Formula of Population Variance in Ungrouped Data:
Where:
Grouped data refers to a dataset where individual data points are grouped or categorized into intervals or classes. Each interval represents a range of values, and the frequency of data points falling within each interval is recorded.
Formula of Population Variance in Grouped Data:
Where:
The table gives the differences between the population variance and sample variance:
Population Variance | Sample Variance |
|---|---|
Population variance is calculated using the entire data set. | Sample variance is calculated using only a sample of the data set. |
You calculate the population variance when the dataset you’re working with, represents an entire population, i.e. every value that you’re interested in. | You calculate the sample variance when the dataset you’re working with represents a a sample taken from a larger population of interest. |
The formula to calculate population variance is: where:
| The formula to calculate sample variance is: where:
|
Key differences between population variance and standard deviation are:
Aspect | Standard Deviation | Population Variance |
|---|---|---|
Definition | Measures the spread of data points in a population from the population mean. | Measures the dispersion of data points in a population from the population mean. |
Formula | ||
Units | Squared units of the original data (e.g., square meters, square dollars). | Same units as the original data (e.g., meters, dollars). |
Bias Correction | Uses N in the denominator. | Uses N−1 in the denominator. |
Representation | σ2 | σ |
Sensitivity to Outliers | Less sensitive, as it squares differences before averaging. | More sensitive, as it considers absolute differences. |
Solution:
Calculate the squared deviations from the mean:
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.
Solution:
Calculate the squared deviations from the mean:
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.
Solution:
Calculate the squared deviations from the mean:
Sum up the squared deviations = 1 + 1 + 0 + 4 + 4 + 1 + 9 = 20
Divide by the total number of observations (which is 7) =
Therefore, the population variance for this temperature data set is approximately 2.86.
Q1. The population variance is also called:
a) Sigma squared
b) Sigma cubed
c) Sigma
d) None of the above
Q2. When a sample variance of 25 is obtained from a sample of 10 items from a normal population, the 80% confidence interval for a population variance is:
a) 12.3 to 57.1
b) 13.3 and 67.7
c) 14.1 to 46.25
d) 15.3 to 53.98
3. The sampling distribution of the ratio of independent sample variances from two normally distributed populations with equal variances is the:
a) Chi-square distribution
b) Normal distribution
c) F distribution
d) T distribution
4. These sample results were obtained for independent random samples from two normally distributed populations. Sample 1: Sample Size 10, Sample Variance 25. Sample 2: Sample Size 16, Sample Variance 20. Using a .05 level of significance, which conclusion would be reached for these data?
a) There is a statistically significant difference between the variances of the two populations.
b) There is no statistically significant difference between the variances of the two populations.
c) Insufficient data - can’t tell in this case.
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