Definition of sample variance:
In 1893 the famous statistician Karl Pearson gave the definition of standard deviation as a measure of dispersion and gave a method for its calculation. Of all measures of dispersion the standard deviation is most commonly used. To understand what is sample variance, let us first see what is sample standard deviation.
Standard deviation: The positive square root of the number, obtained by dividing the sum of the squares of deviations from mean of observations of given data, by the number of observations is called the standard deviation of the data. It is denoted by s. Thus if x ¯ is the mean of n observations x_i (i = 1,2,3,…n), then,
With this background let us now define sample variance:
It is convenient to calculate the standard deviation in two steps. In fact, the number obtained by dividing the sum of the squares of deviations from the mean of observations of the given data by the number of observations is called the variance of the data. The variance is denoted by s^2 or s^2. After calculating the variance, its positive square root gives the standard deviation s. Thus, based on the above sample variance definition, we can write the sample variance formula as follows:
Calculation of variance for ungrouped data:
Direct method: Calculate the mean x ¯ of observations x1, x2, x3, … , x_n. Find the deviation x_i - x ¯ for each x_i. Calculate the square (x_i - x ¯)^2 for each x_i. Find the sum ?(x_i - x ¯)^2. Divide ?(x_i - x ¯)^2 by n to get the variance s^2.
If the values of x_i are relatively small and the mean is a fraction or a lengthy decimal, then we have an alternative formula for variance as follows:
Because of the presence of x_i^2 in the alternate formula for variance, finding variance by this formula could be tedious when x_i^2 are relatively large. The first method is also time consuming if the mean is a fraction (and not an integer). Fortunately, the numbers to be handled can be made small and the fractional mean can be avoided in calculations, by taking deviations of x_i^2 from an appropriately chosen number A, called the assumed mean.
In 1893 the famous statistician Karl Pearson gave the definition of standard deviation as a measure of dispersion and gave a method for its calculation. Of all measures of dispersion the standard deviation is most commonly used. To understand what is sample variance, let us first see what is sample standard deviation.
Standard deviation: The positive square root of the number, obtained by dividing the sum of the squares of deviations from mean of observations of given data, by the number of observations is called the standard deviation of the data. It is denoted by s. Thus if x ¯ is the mean of n observations x_i (i = 1,2,3,…n), then,
It is convenient to calculate the standard deviation in two steps. In fact, the number obtained by dividing the sum of the squares of deviations from the mean of observations of the given data by the number of observations is called the variance of the data. The variance is denoted by s^2 or s^2. After calculating the variance, its positive square root gives the standard deviation s. Thus, based on the above sample variance definition, we can write the sample variance formula as follows:
Direct method: Calculate the mean x ¯ of observations x1, x2, x3, … , x_n. Find the deviation x_i - x ¯ for each x_i. Calculate the square (x_i - x ¯)^2 for each x_i. Find the sum ?(x_i - x ¯)^2. Divide ?(x_i - x ¯)^2 by n to get the variance s^2.
If the values of x_i are relatively small and the mean is a fraction or a lengthy decimal, then we have an alternative formula for variance as follows:
Because of the presence of x_i^2 in the alternate formula for variance, finding variance by this formula could be tedious when x_i^2 are relatively large. The first method is also time consuming if the mean is a fraction (and not an integer). Fortunately, the numbers to be handled can be made small and the fractional mean can be avoided in calculations, by taking deviations of x_i^2 from an appropriately chosen number A, called the assumed mean.
