Sample variance is used to calculate the variability in a given sample. A sample is a set of observations that are pulled from a population and can completely represent it. The sample variance is measured with respect to the mean of the data set. It is also known as the estimated variance. Show
As data can be of two types, grouped and ungrouped, hence, there are two formulas that are available to calculate the sample variance. Furthermore, the square root of the sample variance results in the sample standard deviation. In this article, we will elaborate on sample variance, its formulas, and various examples. What is Sample Variance?Sample variance is used to measure the spread of the data points in a given data set around the mean. All observations of a group are known as the population. When the number of observations start increasing it becomes difficult to calculate the variance of the population. In such a situation, a certain number of observations are picked out that can be used to describe the entire group. This specific set of observations form a sample and the variance so calculated is the sample variance. Sample Variance DefinitionSample variance can be defined as the expectation of the squared difference of data points from the mean of the data set. It is an absolute measure of dispersion and is used to check the deviation of data points with respect to the data's average. Sample Variance ExampleSuppose a data set is given as 3, 21, 98, 17, and 9. The mean (29.6) of the data set is determined. The mean is subtracted from each data point and the summation of the square of the resulting values is taken. This gives 6043.2. To get the sample variance, this number is divided by one less than the total number of observations. Thus, the sample variance is 1510.8. Sample Variance FormulaThere can be two types of data - grouped and ungrouped. When data is in a raw and unorganized form it is known as ungrouped data. When this data is sorted into groups, categories, or tables it is known as grouped data. The sample variance formulas for both types of data are specified below:
n = total number of observations. N = \(\sum_{i=1}^{n} f_{i}\) f = the frequency of occurrence of an observation for grouped data \(m_{i}\) = Mid-point of the ith interval Mean for grouped data, \(\overline{x}\) = \(\frac{\sum_{i=1}^{n} m_{i}f_{i}}{\sum_{i=1}^{n} f_{i}}\) Mean for ungrouped data, \(\mu = \frac{\sum_{i=1}^{n}x_{i}}{n}\) The sample variance, on average, is equal to the population variance. Let us understand the sample variance formula with the help of an example. Example: There are 45 students in a class. 5 students were randomly selected from this class and their heights (in cm) were recorded as follows:
Sample size (n) = 5 Sample Mean = (131 + 148 + 139 + 142 +152 ) / 5 = 712 / 5 = 142.4 cm Using the sample variance formula, Sample Variance =\(\frac{\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{n-1}\) = \(\frac{\sum_{i=1}^{5}(x_{i}-142.4)^{2}}{5-1}\) = [(131−142.4)2+(148−142.4)2+(139−142.4)2+(142−142.4)2+(152−142.4)2] / 4 = 66.3 cm2 Answer: Sample Mean = 142.4 cm, Sample Variance = 66.3 cm2. How to Calculate Sample Variance?Depending upon the type of data available, there can be different steps that can be used to calculate the sample variance. However, the general algorithm that should be followed is given below: Suppose the data set is given as {5, 6, 1}
Sample Variance vs Population VarianceBoth sample variance and population variance are used to measure how far a data point is from the mean of the data set. However, the value of the sample variance is higher than the population variance. The table given below outlines the difference between sample variance and population variance.
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Important Notes on Sample Variance
FAQs on Sample VarianceWhat is the Sample Variance in Statistics?In statistics, sample variance is calculated on the basis of sample data and is used to determine the deviation of data points from the mean. What is the Symbol of Sample Variance?Variance is usually represented using sigma square that is written as \(\sigma ^{2}\). However, to avoid confusion between population and sample variance, the latter is represented as s2. What is the Formula for Sample Variance?The formulas for sample variance are given as follows:
How to Find the Sample Variance?The steps to find the sample variance are as follows:
Can Sample Variance be Negative?No, the sample variance can never be negative. The sample variance is the square of the deviation from the mean. As a value resulting from a square can never be negative, thus, sample variance cannot be negative. Is Sample Variance the Same as Standard Deviation?The square root of the sample variance will result in the standard deviation. The unit of measurement of the sample variance will be different as compared to the data while the unit of the sample standard deviation will be the same. What is the Difference Between Sample Variance and Population Variance?The variance that is calculated using the sample data gives the sample variance while the population data gives population variance. The formula for sample variance is \(\frac{\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{n-1}\) and population variance is \(\frac{\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{n}\) What do Small and Big Variance Mean in Sample Variance Formula?A small variance obtained using the sample variance formula indicates that the data points are close to the mean and to each other. A big variance indicates that the data values are spread out from the mean, and from one another. Why is the formula for sample variance different from the formula for population variance?Differences Between Population Variance and Sample Variance
The only differences in the way the sample variance is calculated is that the sample mean is used, the deviations is summed up over the sample, and the sum is divided by n-1 (Why use n-1?).
How does the formula for the sample mean differ from the formula for population mean?How does the formula for the sample mean differ from the formula for population mean? The Greek letter mu, *u*, is used to represent the population while, *X*, x-bar is used for the sample mean. The formulas are functionally the same, but 'n' (the sample size) is used instead of 'N' (the population size).
Is there a difference between the variance of the population and variance of the sampling distribution of the sample means?“That is, the variance of the sampling distribution of the mean is the population variance divided by N, the sample size (the number of scores used to compute a mean). Thus, the larger the sample size, the smaller the variance of the sampling distribution of the mean.
Is variance and population variance the same?Definition. The variance defines a measure of the spread or dispersion within a set of data. There are two types: the population variance, usually denoted by σ2 and the sample variance is usually denoted by s2 .
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