The document provides information about measures of central tendency (mean, median, mode) and measures of dispersion (range, quartiles, variance, standard deviation) using examples of data distributions. It defines key terms like mean, median, mode, range, quartiles, variance and standard deviation. It also shows how to calculate and interpret these measures of central tendency and dispersion using sample data sets.

1. Math 3 Warm up
Below is a histogram of the ages of people in a family.
Find the Mean, Median, and Mode of the data

2. Math 3 Warm up
Below is a histogram of the ages of people in a family.
Find the Mean, Median, and Mode of the data
Score Distribution:
20: 20 20
30: 30 30 30 30
40: 40 40 40 40
50: 50 50 50 50 50
60: 60 60 60
70: 70
90: 90
Mean: 45.5
Median: 45
Mode: 50

3. Summarizing Distributions
Two key characteristics of a frequency distribution
Measures of Central Tendency (Mean, Median, Mode)
What is in the “Middle”?
What is most common?
What would we use to predict?
Measures of Dispersion (Spread)
How Spread out is the distribution?
What Shape is it?

5. Voter Turnout in 50 States - 1940

6. Mean

7. 7
Measures of Dispersion
There are three main measures of
dispersion:
The range
Quartiles
Variance / standard deviation

8. 8
The Range
The range is defined as the difference
between the largest score in the set of data
and the smallest score in the set of data.
What is the range of the following data:
4 8 1 6 6 2 9 3 6 9
The largest score is 9; the smallest score is
1; the range is : 9 - 1 = 8

9. Find the range from this stem and leaf plot.

10. 10
When To Use the Range
The range is used when
You are presenting your results to people with
little or no knowledge of statistics
The range is rarely used in scientific work
as it is fairly insensitive.
It depends on only two scores in the set of data,
First and last.
Two very different sets of data can have the
same range:
1 1 1 1 9 vs 1 3 5 7 9

11. 11
Quartiles
The first quartile is the 25% mark
The median is the 50 % mark
The third quartile is the 75% mark

12. Litter Size 2 3 4 5 6
Number of
Litters
1 6 8 11 1
The table below displays a frequency distribution
showing a cat breeder’s records for the number of
kittens born in a certain year..

13. Kitten Data
2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6
Lower half Upper half
First Quartile: 3
median of lower half
Third Quartile: 5
median of upper half
Median: 4
Second Quartile
You know that the median of a data set divides the
data into a lower half and an upper half. The median
of the lower half is the lower quartile, and the
median of the upper half is the upper quartile.

14. Kitten Data
2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 6
Lower half Upper half
First quartile: 3
median of lower half
Third quartile: 5
median of upper half
Median
Second Quartile:
4
This can be displayed in a diagram known as a
Box and Whisker plot.

15. 23 24 26 29 31 31 33 35
Use the given data to make a box-and-whisker plot.
31, 23, 33, 35, 26, 24, 31, 29
Example
Step 1. Order the data from least to greatest. Then
find the Median, first quartile, third quartile, minimum
and maximum.

16. Use the given data to make a box-and-whisker plot.
22 24 26 28 30 32 34 36 38
Step 2. Draw a number line and plot a point above
each value.
23 24 26 29 31 31 33 35
Example Continued

17. 17
Step 2. Draw a number line and plot a point above
each value.
Use the given data to make a box-and-whisker plot.
22 24 26 28 30 32 34 36 38
Step 3. Draw the box and whiskers.
23 24 26 29 31 31 33 35
Example Continued

18. Use the following data to make a box-and-
whisker plot.
91, 87, 98, 93, 89, 78, 94
78 87 91 94 98

19. 19
What is Variance?
The larger the variance is, the more the
scores deviate, on average, away from the
mean
The smaller the variance is, the less the
scores deviate, on average, from the mean

20. 20
Variance (Var.)
Variance is defined as the average of the
square deviations:
 
N
X
2
2  




21. 21
What Does the Variance Formula
Mean?
First, it says to find the mean.
Then subtract the mean from each of the
data points
These differences are called a deviates or
deviations from the mean
Deviation = data - mean

22. Example
Here are the results of a history test:
Mean = 79
22
We don’t want
negative deviations, so
we “square them” to
make them positive!

23. Example
Here are the results of a history test:
Mean = 79
23
Squares of
Deviations
_____________

24. Variance is the mean of the squared
deviation scores
Squares of
Deviations
_____________
225
100
49
9
16
49
64
256
Mean = 79 Variance =
Mean of Dev2
_____________

25. Recap –
How to find Variance (Var)
To find Variance:
Find the mean
Find the deviations from the mean
Square the deviations (to make them positive)
Find the mean of the squares
25

26. 26
Standard Deviation (Std.)
Standard deviation = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
It follows then that:
Variance = standard deviation2

27. 27
Standard Deviation
When the deviate scores are squared in variance,
their unit of measure is squared as well
E.g. If people’s weights are measured in pounds,
then the variance of the weights would be expressed
in pounds2 (or squared pounds)
Since squared units of measure are often
awkward to deal with, the square root of variance
is often used instead
The standard deviation is the square root of variance

28. Variance is the mean of the squared
deviation scores
Standard Deviation is the square root of
variance
Squares of
Deviations
_____________
225
100
49
9
16
49
64
256
Mean = 79
Variance =
Mean of Dev2
_____________
96
Standard
Deviation =
_____________
96 = 9.8
𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒

29. Practice
Find the mean, variance, and standard
deviation (To the nearest tenth)
29
Data Mean Deviation Deviations2 Variance Standard
Deviation
1
4
6
6
7
8
8