Variance and standard deviation are the most commonly used measures of variability.
Sample variance is denoted by s2, and population variance by σ2.
Sample standard deviation is denoted by s, and the parameter by σ.
Variance and standard deviation are directly related, because variance is the square of the standard deviation, and standard deviation is the square root of the variance.
Like the arithmetic mean, they are used on data that are at the interval or ratio level of measurement.
Variance and standard deviation are used in inferential statistics.
The formal definition of variance is that it is the average of the square of the difference in scores and arithmetic mean. From the definition it follows that when calculating the variance, the difference of each result from the arithmetic mean is first calculated, then the difference is squared and the squared difference is divided by the number of results in the distribution. The formula for calculating the variance of a population is
The formula for calculating the variance of a sample, if used for the purpose of description of data is
and if used for statistical inference, then the formula is
There are two formulas for calculating sample variance, depending on the purpose for which they are used. The difference between the two formulas is in the denominator: the formula for calculating sample variance used for the purpose of describing data has n in the denominator, and the formula used to calculate sample variance for statistical inference has n-1 in the denominator. This means that as n, or the sample size, increases, the difference between the variance values calculated using the two formulas will decrease.
Since the variance value calculated using the data description formula is always smaller than the variance value calculated using the statistical inference formula, the variance value calculated using the first formula will be underestimated compared to the population variance. The smaller the sample, the more the calculated sample variance value will be underestimated compared to the true population variance value. For this reason, the formula that has n-1 in the denominator is most often used to calculate the sample variance.
The formula for calculating the standard deviation of a population is
a for a sample, regardless of the purpose of use
Variance is not suitable for describing variability because it contains the square of the deviation of individual results from the arithmetic mean. Therefore, standard deviation is used to describe variability.
Example: Calculating variance and standard deviation
For the example of the height of girls in the 52 class of the "X" primary school (in cm) that we used for the example of calculating the arithmetic mean, median, mode and rank, we obtained the following data: 140, 141, 138, 140, 122, 160, 154, 132, 148, 135, 140. We calculated that the arithmetic mean value is 140.91 cm.
If we were only interested in the height of girls in the 52 category, then we would be dealing with a population. In this example, we will consider the data listed as a sample.
We will show the procedure for calculating the variance and standard deviation through a table that was formed taking into account the following formula for sample variance:
| X | Xi-X | (Xi-X)2 | X2 |
| 140 | -0.91 | 0.8281 | 19600 |
| 141 | 0.09 | 0.0081 | 19881 |
| 138 | -2.91 | 8.4681 | 19044 |
| 140 | -0.91 | 0.8281 | 19600 |
| 122 | -18.91 | 357.5881 | 14884 |
| 160 | 19.09 | 364.4281 | 25600 |
| 154 | 13.09 | 171.3481 | 23716 |
| 132 | -8.91 | 79.3881 | 17424 |
| 148 | 7.09 | 50.2681 | 21904 |
| 135 | -5.91 | 34.9281 | 18225 |
| 140 | -0.91 | 0.8281 | 19600 |
| Σ=1550 | 0 | Σ=1068.909 | Σ=219478 |
The sum of the squares of the deviations of the results from the arithmetic mean is 1068.909, s2=1068.909/(11-1)=106.89, and s=√106.89=10.34
Answer: The variance of the height of girls in 52 class of the „X“ school is s2=106.89, and the standard deviation s=10.34.
The value of the standard deviation or variance can never be a negative number. A negative value of the standard deviation or variance indicates an error in the calculation.
The smallest possible value of the variance and standard deviation is 0 and this occurs when all the results in the distribution are equal.
The variance and standard deviation are sensitive to extreme values, because they are based on the distance of individual results from the arithmetic mean.
variability
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