Don't Let the Formula Scare You!
In my opinion, if the "powers that be" really wanted to entice kids into embracing mathematics, they would use kittens, puppies, bunny rabbits, etc., as math symbols, not Greek (geek) letters.
Want to scare a math-challenged student into closing the math textbook so fast that it blows out the candles in the room? Show them this:
| σ2 = |
| ||||
| N | |||||
| σ = √ | σ2 | ||||
That's the scary-looking formula for calculating the variance (σ2) and standard deviation (σ) of a population data set.
But wait! Keep your candles burning!
As you will see in the example further down the page, the steps to calculating variance and standard deviation are much easier than trying to decipher the Greek/geek formula.
What is Standard Deviation?
Standard Deviation is simply one of several methods for summarizing the dispersion of the values in a set of data.
Specifically, standard deviation is the square root of the variance, which attempts to summarize the variability or dispersion of values relative to the mean of the entire set.
A small standard deviation indicates the values are tightly grouped around the mean (average) of the data set.
A large standard deviation indicates the values are not tightly grouped around the mean (average) of the data set.
Population Vs Sample Statistics
Population: The data set is the total set of elements of interest for a given problem. Population parameters in the formulas on this page are denoted by σ and μ.
Sample: The data set represents only a fraction of the population as defined above. Sample parameters in the formulas on this page are denoted by s and X.
4 Simple Steps
As I stated earlier, calculating standard deviation is much easier than the formula depicts. Here are the 4 simple steps:
- Find the mean of the data set.
- Find the sum of the squared differences from the mean.
- Divide the result in step #2 by n (population) or n - 1 (sample), where n is the number of items in the set.
- Find the square root of the result in step #3.
To see just how easy the above steps are, let me walk you through an example.
Example Problem
To illustrate how easy it is to calculate variance and standard deviation, I will use the following data set:
Note: Decimals in this example are rounded to 2 places before they are displayed on the screen.
Step #1: Calculate the mean
The mean is the average of all numbers in a data set. To calculate the mean of a set of numbers, you add all of the items together and then divide that result by the number of items within the set.
| Data set: 5, 4, 7, 9, 6, 8, 7, 5, 4, 5 |
| Data set contains 10 items |
| Mean = (5 + 4 + 7 + 9 + 6 + 8 + 7 + 5 + 4) ÷ 10 |
| Mean = 60 ÷ 10 |
| Mean = 6 |
Step #2: Find the sum of the squared distances from the mean.
For each item in the data set we: (1) subtract the mean of the entire set from the item, (2) square the result, and then (3) sum all of the squared results, like this:
| Cnt n | X | X - μ | (X - μ)2 |
|---|---|---|---|
| 1 | 5 | 5 - 6 = -1 | (-1)2 = 1 |
| 2 | 4 | 4 - 6 = -2 | (-2)2 = 4 |
| 3 | 7 | 7 - 6 = 1 | (1)2 = 1 |
| 4 | 9 | 9 - 6 = 3 | (3)2 = 9 |
| 5 | 6 | 6 - 6 = 0 | (0)2 = 0 |
| 6 | 8 | 8 - 6 = 2 | (2)2 = 4 |
| 7 | 7 | 7 - 6 = 1 | (1)2 = 1 |
| 8 | 5 | 5 - 6 = -1 | (-1)2 = 1 |
| 9 | 4 | 4 - 6 = -2 | (-2)2 = 4 |
| 10 | 5 | 5 - 6 = -1 | (-1)2 = 1 |
| n = 10 | Sum = 60 | Σ (X - μ)2 = 26 |
Step #3: Calculate the variance.
This step depends on whether or not the set you are working with represents the total data (population) or partial data (sample). Here is how you calculate the variance from the results in step #2 for either case (μ and X are both symbols for Mean, with the only real difference indicated by red text):
| Population Variance (σ2) | ||||
|---|---|---|---|---|
| σ2 = | Σ (X - μ)2 | = | 26 | = 2.60 |
| n | 10 | |||
| Sample Variance (s2) | ||||
|---|---|---|---|---|
| s2 = | Σ (X - X)2 | = | 26 | = 2.89 |
| n - 1 | 9 | |||
Step #4: Calculate the square root of the variance.
This step is basically the same for both the population and the sample standard deviation calculations. The only difference is that the Variance was computed differently in Step #3 (n in denominator vs n - 1):
| Population Standard Deviation (σ) | ||||
|---|---|---|---|---|
| σ = √ | σ2 | = √ | 2.60 | = 1.61 |
| Sample Standard Deviation (s) | ||||
|---|---|---|---|---|
| s = √ | s2 | = √ | 2.89 | = 1.70 |
See how easy that was?