http://www.ck12.org Chapter 5. The Shape, Center and Spread of a Normal Distribution - Basic
5.2 Calculating the Standard Deviation
Learning Objectives
- Understand the meaning of standard deviation.
- Understanding the percents associated with standard deviation.
- Calculate the standard deviation for a normally distributed random variable.
Introduction
You have recently received your mark from a recent Math test that you had written. Your mark is 71 and you are
curious to find out how your grade compares to that of the rest of the class. Your teacher has decided to let you
figure this out for yourself. She tells you that the marks were normally distributed and provides you with a list of the
marks. These marks are in no particular order –they are random.
32 88 44 40 92 72 36 48 76
92 44 48 96 80 72 36 64 64
60 56 48 52 56 60 64 68 68
64 60 56 52 56 60 60 64 68
We will discover how your grade compares to the others in your class later in the lesson.
Standard Deviation
In the previous lesson you learned that standard deviation was the spread of the data away from the mean of a set of
data. You also learned that 68% of the data lies within the two inflection points. In other words, 68% of the data is
within one step to the right and one step to the left of the mean of the data. What does it mean if your mark is not
within one step? Let’s investigate this further. Below is a picture that represents the mean of the data and six steps
–three to the left and three to the right.
These rectangles represent tiles on a floor and you are standing on the middle tile –the blue one. You are then asked
to move off your tile and onto the next tile. You could move to the green tile on the left or to the green tile on the
right. Whichever way you move, you have to take one step. The same would occur if you were asked to move to the
second tile. You would have to take two steps to the right or two steps to the left to stand on the red tile. Finally, to
stand on the purple tile would require you to take three steps to the right or three steps to the left.
If this process is applied to standard deviation, then one step to the right or one step to the left is considered one
standard deviation away from the mean. Two steps to the left or two steps to the right are considered two standard
deviations away from the mean. Likewise, three steps to the left or three steps to the right are considered three