1. Statistics and Probability
CLAST Review Workshop
Statistics and Probability
3. Measures of Central Tendency
(skill I D 2)

2. Mean
Median
Mode
The
3Ms
Dr. Carmen Arteaga 2

3. Central Tendency
A distribution of statistics that groups about an apparent center may be described conveniently by that central value. For example, say that you have to take a statistics course and you learn that there are only two professors who teach it. One gives exams that average 73; the other gives exams that average 78. Who would you choose as your professor?
Mean – the “average” of all the data
Median – the middle datum
Mode – the most often occurring datum
Dave Saha, Kathy Lyden 3

4. Arithmetic Mean Summary
Given a set of numbers, xi:
Get the count, the number of numbers in the set. count = n
Sum all the numbers  xi = x1 + x2 + x3 +  + xn
Divide the sum by the count  xi x1 + x2 + x3 +  + xn
mean = x = =
n n
Kathy Lyden, Dave Saha 5

5. Arithmetic Mean 3 2 5 3 6 9 3 4 9 3 3+2+5+3+6+9+3+4+9+3 = 47
Given this set of 10 numbers ---
To find their mean, first sum the numbers ---
= 47
then divide the sum by the count (10) ---
47 ÷ 10 = 4.7
Dr. Carmen Arteaga 4

6. Median Summary Given a set of numbers, xi:
Sort the numbers in the set in ascending order
If the count is odd, the median is the middle number.
If the count is even, the median is the mean of the two middle numbers.
Kathy Lyden, Dave Saha 8

7. Median with n Odd
To find the median of this set of numbers ---
Sort the numbers in ascending order
The median is the middle number.
There are an equal number of entries above and below the median.
Dr. Carmen Arteaga 6

8. Median with n Even
To find the median of this set of numbers ---
Sort the numbers in ascending order

The median is the average of the two middle numbers.
There are an equal number of entries above and below the median.
( )(4 + 5) = 4.5
Dr. Carmen Arteaga, Dave Saha 7

9. Mode Summary Given a set of numbers, xi:
The mode is the most frequently occurring number in the list (it may help to sort the list).
Multiple modes may occur. Example: The following data is bimodal: , 10, 10, 10, 20, 25, 25, 40, 40, 40, 40, 55, 70 The two modes are the values 10 and 40, each of which occurs four times.
If every value in the data occurs with equal frequency, there is no mode Example: 10, 10, 20, 20, 30, 30, 40, 40, 50, 50
Dave Saha 10

10. 2 3 3 3 4 5 6 9 9 Mode To find the mode of this set of numbers
The mode is the most frequent value in the list. In some data sets there may be more than one mode, or no mode at all.
Dr. Carmen Arteaga, revised 9

11. Property Values in my neighborhood Count = 23 $450,000 450,000 400,000
250,000
125,000
100,000
70,000
Property Values
in my
neighborhood
Count = 23
Dr. Carmen Arteaga 11

12. Mean Sum of property values = $3,610,000 Average (Mean) value =
$450,000
450,000
400,000
250,000
125,000
100,000
70,000
Mean
Sum of property values = $3,610,000
Average (Mean) value =
$3,610,000 ÷ 23 = $156,956.52
Dr. Carmen Arteaga 12

13. Median value is the value
$450,000
450,000
400,000
250,000
125,000
100,000
70,000
Median
Median value is the value
of house “number 12”
$100,000
Dr. Carmen Arteaga 13

14. $450,000
450,000
400,000
250,000
125,000
100,000
70,000
Mode
In the neighborhood, there are more houses valued at $70,000 than at any other price
Dr. Carmen Arteaga 14

15. Property Values in my neighborhood
$450,000
450,000
400,000
250,000
125,000
100,000
70,000
Property Values in my neighborhood
Mean value = $156,956.52
Median value = $100,000
Mode value = $ 70,000
Dr. Carmen Arteaga 15

16. HOUSE VALUE DISTRIBUTION
Dr. Carmen Arteaga 17

17. Which value should be used?
Mean value = $156,956.52
Median value = $100,000
Mode value = $ 70,000
Average is represented by the mean value.
The value used should be reported.
Dr. Carmen Arteaga 16

18. General Rule for Choosing:
Use the Mode only if the mean and median do not make sense or are not helpful.
Use the Median if the graph’s shape is too skewed or if the data contains outliers.
Use the Mean if the data is near bell-shaped.
NOTE: If the data is bell-shaped, then the mean, median, and mode are all equal to each other.
Dr. Carmen Arteaga 18

19. Same Average, Different Picture
Midterm exam scores for two classes:
Class 1: 25, 40, 55, 65, 100, 100, 100
mean = median = 65
Class 2: 65, 65, 65, 65, 75, 75, 75
The mean and/or median most fairly represent
the scores for Class 2.
The information on the following slides is above and beyond the
CLAST level. Several terms have not been defined.
Dr. Carmen Arteaga, revised 19

20. Variation Will Tell the Rest of the Story
Class 1: Range = Standard dev. = 31.28
Class 2: Range = Standard dev. = 5.35
The more range in the data, the less representative are the mean, median, and mode.
Dr. Carmen Arteaga 20

21. Empirical Rule If the data is near bell-shaped, then approximately:
68% of the values are between
mean - std. dev. and mean + std. dev.
95% of the values are between:
mean - 2 std. dev. and mean + 2 std. dev.
99.7% of the values are between:
mean - 3 std. dev. and mean + 3 std. dev.
Dr. Carmen Arteaga 21

22. THE FIVE NUMBERS SUMMARY
Minimum (Mi) and Maximum (Ma)
Median (Me)
Lower Quartile (LQ) and Upper Quartile (UQ)
25%
25%
25%
25%
Mi LQ Me UQ Ma
Dr. Carmen Arteaga 22

23. Finding Quartiles (Part 1)
Step 1. Find the median
Lower Half Upper Half 4
Dr. Carmen Arteaga, revised 23

24. Finding Quartiles (Part 2)4
Lower Half Upper Half
Step 2. Find the median of each half
7.5
2.5 UQ LQ
Dr. Carmen Arteaga, revised 24