Standard Deviation

Standard deviation is part of statistics and studies how data deviates or changes from the data’s mean (average). Basically, it looks how the data spreads out in increments from the mean—a low standard deviation means that most of the numbers are close to the average, and a high standard deviation means that the numbers are more spread out.

For example, imagine a teacher gives her 26 students a pre-assessment for a math class to learn how prepared her students are. If she scores all the pre-assessments and calculates the standard deviation of the scores, what would the result tell her? If the test scores have a low standard deviation, it means most of the students have a similar knowledge base, and it would be easier to tailor her curriculum.

However, if the standard deviation is very high, it means there is a large variance in what the students know, and she may have to consider splitting the students into two groups based on knowledge level.  

Calculating Standard Deviation

The formula to calculate standard deviation is:

To calculate standard deviation, follow these steps:

1. Find the mean of the values in the population
2. Subtract the mean from each value in the population
3. Square each difference
4. Find the sum of the squared differences
5. Divide the sum of the squared differences by the number of population values in the population (mean)
6. Take the square root of the final mean to find the standard deviation

Population vs. Sample

The above equation is how you calculate standard deviation for a population. However, if the data is a sample (a selection from the total population), then the calculation changes. 

Still, the steps remain very similar.

To calculate sample standard deviation:

1. Find the mean of the sample values
2. Subtract the mean from each sample value 
3. Square each difference
4. Find the sum of the squared differences
5. Divide the sum of the squared differences by the number of sample values minus one (this is the only step that differs!)
6. Take the square root of the above result to find the sample standard deviation 

Examine the following practice problems to compare the population standard deviation and the sample standard deviation.

Take the following data set: 

1, 2, 4, 5, 8

Let’s first assume that these values comprise the total population.

Now, here’s how to solve the problem if these values are just a sample of the total population.

We can see that the values are slightly different. In particular, the sample standard deviation is slightly larger than the population standard deviation.

If you are only working with a sample, you are less confident of the overall variation, thus we choose to estimate that the standard deviation is a bit higher.

Standard Deviation Practice Problems

Answers to Practice Problems

1. Mean = $3,995.24. Standard deviation = 264.53.
2. Set B has a larger standard deviation.
3. When this group is treated as a population, standard deviation = 6.02. As a sample, standard deviation = 6.25.
4. D
5. A