Mean and Standard Deviation Notes

• When describing a set of data, it’s often useful to be able to talk about roughly where the data is

centered and how much the data varies or is ____________ out.spread

Measures of Center

• There are several useful measures of center:• ___________• ___________• and mode.

• The one we use most often is the mean (i.e. the arithmetic average).

• We use ______ or ______ to symbolize mean.• Practice: Calculate the mean of the data set: Z={1, 2, 5, 7, 9}

meanmedian

   

       

Measures of Spread

• There are also several measures of how much the data are spread out: range, quartiles, IQR, etc. However, the measure that is used most often is more difficult to calculate, but is based on a very solid idea.

• The variance, just like the name suggests, is a measure of how much the data _______. The variance, and it’s more useful counterpart, the standard _________, are the most used way to describe how much a data set are spread out.

varydeviation

Example 1

• Let’s calculate the variance for the data set: Z={1, 2, 5, 7, 9}

• What we would like to measure is how much the data vary from the center (i.e. the mean) of the data set. In other words, what is the difference between each data point and the mean?• These are called ____________ from the mean.deviations

 1-4.8=

-3.8

Z 1 2 5 7 9

2-4.8=

-2.8

5-4.8=

0.2

7-4.8=

2.2

9-4.8=

4.2

Example 1 cont.

• Now that we have how much each one varies, how much do the data vary in total? _______

• Would this happen with every data set? _______• Finding the difference between each data point and the mean is a good way to measure how much the data vary, but the total of how they vary will always be _____. How can we adjust what we’ve done to make this a more useful measure?• Hint: How can we make all of our numbers positive?

0

Yes

0

Absolute Value Square them all

Example 1 cont.

• Let’s take the best suggestion and see what we get. These are called ____________ deviations about the mean.

• Now, how can we find the average amount the data vary?

• What is the average amount the data vary?

squared 

 

 

 

Example 1 cont.

• The answer we just found measures how much the data vary and is called the ___________. There are a few problems with it.• First, the numbers are bigger than they were to begin with (because we

squared them). • Second, if we were dealing with a data set that had labels, like “apples” for

instance, the units on what we just found would be apples ____________, not apples.

• How can we adjust what we found above to make it a more useful measure with the correct units? _____________

• Now, what is the best measure for the average amount the data deviates from the mean?

variance

squared

Square root

 

Example 1: Final Answer

• So, for the data set Z={1, 2, 5, 7, 9}, the measure of center and spread we’ll use is:• Mean = ________ -and-• Standard Deviation = _________

4.8

2.99

Let’s Put it all Together

•  

   

Example 2

• Compute the mean and standard deviation for the data set X={1, 2, 3, 3, 4, 6, 6, 6, 6}. 

1 2 3 3 4 6 6 6 6

1-4.111=-3.111

2-4.111=-2.111

3-4.111=-1.111

3-4.111=-1.111

4-4.111=-0.111

6-4.111=1.889

6-4.111=1.889

6-4.111=1.889

6-4.111=1.889

  

 Variance

 Standard Deviation

Example 3

• Consider the following data set Y={20, 20, 21, 21, 21, 23, 184} and think about these two questions:• 1) Where is the center? 2) How spread out is this data? More or less than the last example?

   

Final Note

• Greek letters are often used in statistics.

• The mean is represented by the Greek letter _____ (____) and the standard deviation is represented by the Greek letter ______ (_______).

• If we have just a sample of the data (not the entire population), then we use ____ (_______) for the mean and _____ for the standard deviation of that sample.

• Please note: the formula for the standard deviation of a sample is slightly different than finding the standard deviation of an entire population.

 

 

sigma

  s