An Introduction to Average Values and Standard Deviation
Tim Sundstrom
Average values and standard deviation are used a lot in our industry, and it is easy to calculate them using Excel.
Assume you have 20 people, and you want to find their average height and the standard deviation. Create a list of the people, along with their heights.
Most of you know that to find the average height of this group, you add these heights together and divide the sum by 20. In this case, the average value is 172.4 cm. You can also plot their heights on a graph, which shows a dotted line at the average height of 172.4.
Figure 1: Group of 20 people with an average height of 172.4 cm.
But what if you have a group with more variation in height? In this next group, as shown in Figure 2, the average height is 168 cm, but the deviations are higher. It is also possible to have a group with low deviations. In any case, you need a large group to see accurate and realistic values.
Figure 2: Group of people with an average height of 168 cm.
But what is missing? There is a parameter called standard deviation which is used to describe data sets’ deviation from the average value. The graph in Figure 1 shows a normal deviation, and the graph in Figure 2 shows a high deviation.
Let’s calculate the standard deviation using the group shown in Figure 1. Person 1 is 163.2 cm. If you subtract 172.4 from 163.2, you get -9.2. And that’s the same as the distance from this person’s height to the dotted line in the graph. Person 2 is 190 cm; subtract 172.4, and the difference is 17.6.
If you subtract the average height from the height of each person, you will get different numbers. If the person’s height is above the average value, the difference will be positive; if it is below the average value, the difference will be negative.
Figure 3: Calculate the differences.
Next, square the difference; for example, -9.2 X -9.2 = 84.64, or 84.7 when rounded up. Notice that even though the difference was a negative number, the square is a positive number. All of the squares will be positive.
Figure 4: Square the differences.
Now, add all of these squares together, divide the sum by 20, and take the square root of this number. This will give you the standard deviation, or SD. The standard deviation for this group of people is 9.5 cm.
Figure 5: An equation to find the SD.
You don’t need to actually do this, as your software will do it for you, but it’s good to understand what standard deviation means.
If we use a different group, you can see in the graph that their heights are closer to the average, and the standard deviation is 4 cm.
Figure 6: A group with lower SD.
And if we take a group of people with more varied heights, the standard deviation is higher, in this case, 14.7 cm.
Figure 7: A group with higher SD.
So if you know the standard deviation, you will have an idea of the variation in the data set. Average values and standard deviations are statistical programs that are used a lot in AI systems, such as SPM Instrument’s Decision Support System (DSS) Condmaster Ruby.