Differences Between Population and Sample Standard Deviations

When considering standard deviations, it may come as a surprise that there are actually two that can be considered. There is a population standard deviation and there is a sample standard deviation. We will distinguish between the two of these and highlight their differences.

Qualitative Differences

Although both standard deviations measure variability, there are differences between a population and a sample standard deviation. The first has to do with the distinction between statistics and parameters. The population standard deviation is a parameter, which is a fixed value calculated from every individual in the population.

A sample standard deviation is a statistic. This means that it is calculated from only some of the individuals in a population. Since the sample standard deviation depends upon the sample, it has greater variability. Thus the standard deviation of the sample is greater than that of the population.

Quantitative Difference

We will see how these two types of standard deviations are different from one another numerically. To do this we consider the formulas for both the sample standard deviation and the population standard deviation.

The formulas to calculate both of these standard deviations are nearly identical:

1. Calculate the mean.
2. Subtract the mean from each value to obtain deviations from the mean.
3. Square each of the deviations.
4. Add together all of these squared deviations.

Now the calculation of these standard deviations differs:

• If we are calculating the population standard deviation, then we divide by n, the number of data values.
• If we are calculating the sample standard deviation, then we divide by n -1, one less than the number of data values.

The final step, in either of the two cases that we are considering, is to take the square root of the quotient from the previous step.

The larger the value of n is, the closer that the population and sample standard deviations will be.

Example Calculation

To compare these two calculations, we will start with the same data set:

1, 2, 4, 5, 8

We next carry out all of the steps that are common to both calculations. Following this out calculations will diverge from one another and we will distinguish between the population and sample standard deviations.

The mean is (1 + 2 + 4 + 5 + 8) / 5 = 20/5 =4.

The deviations are found by subtracting the mean from each value:

• 1 - 4 = -3
• 2 - 4 = -2
• 4 - 4 = 0
• 5 - 4 = 1
• 8 - 4 = 4.

The deviations squared are as follows:

• (-3)² = 9
• (-2)² = 4
• 0² = 0
• 1² = 1
• 4² = 16

We now add these squared deviations and see that their sum is 9 + 4 + 0 + 1 + 16 = 30.

In our first calculation, we will treat our data as if it is the entire population. We divide by the number of data points, which is five. This means that the population variance is 30/5 = 6. The population standard deviation is the square root of 6. This is approximately 2.4495.

In our second calculation, we will treat our data as if it is a sample and not the entire population. We divide by one less than the number of data points. So, in this case, we divide by four. This means that the sample variance is 30/4 = 7.5. The sample standard deviation is the square root of 7.5. This is approximately 2.7386.

It is very evident from this example that there is a difference between the population and sample standard deviations.