Name:_____________________________
Date:________________________
Measures of Central Tendency/Variation
A measure of central tendency is a single value that attempts to describe a set of data by
identifying the central position within that set of data.
Mean - The mean is equal to the sum of all the values in the data set divided by the number of
values in the data set.
Median - The median is the middle score for a set of data that has been arranged in order of
lowest to highest.
Mode - The mode is the most frequent score in our data set.
Range - The difference between the lowest and highest values.
Example 1 - The weekly salaries of six employees at McDonalds are $140, $220, $90, $180,
$140, $200. For these six salaries, find: (a) The mean (b) The median (c) The mode (d) The
range.
Example 2 – A student received the following test scores. 40, 61, 95, 79, 50, 80, 109, 42.
For these test scores, find: (a) The mean (b) The median (c) The mode (d) The range.
Example 3 – The frequency table shows the number of job offers received by each student
within two months of graduating with a mathematics degree from a small college.
Find: (a) The mean (b) The median (c) The mode (d) The range for the job offers per student.
Job Offers
0
1
2
Students
2
2
4
Frequency – A measure of the number of times that an event occurs.
3
5
4
2
Name:_____________________________
Date:________________________
Example 4 – The frequency table shows the test grades for each student.
Find: (a) The mean (b) The median (c) The mode (d) The range for the job offers per student.
Test Grades
Students
60
3
65
5
70
5
75
4
80
4
85
2
Measures of Variation - Used to describe the distribution of the data (how spread out data is).
To understand the distribution or dispersion of data, researchers must document the range of
scores or how variable the scores are. The two most frequently used measures of variability are
range and standard deviation.
Standard Deviation and Variance are measures of distance of every score to the mean.
Standard Deviation - The Standard Deviation is a measure of how spread out numbers are.
Variance - The average of the squared differences from the mean.
Mean 𝑥̅ , To find the mean
To find the variance σ2 =
𝐴𝑑𝑑 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠
∑(𝑥− 𝑥̅ )2
𝑛
Standard Deviation - σ, To find the standard deviation σ = √σ2
Example - What are the mean, variance, and standard deviation of these values?
6.9, 8.7, 7.6, 4.8, 9.0
x
𝑥̅
52, 65, 63, 77, 82
(x - 𝑥̅ )
Sum:
(x - 𝑥̅ )2
x
𝑥̅
(x - 𝑥̅ )
Sum:
(x - 𝑥̅ )2
Name:_____________________________
Date:________________________
Use the sums to find the variance and standard deviation.
How to find Standard Deviation in your calculator
1.
2.
3.
4.
Press STAT EDIT (insert values into L1)
Press STAT CALC 1-VAR STAT
Calculate
𝑥̅ is the value for the mean, σx is the standard deviation.
Example 5 – What is the standard deviation of the data sets below?
a.
b.
c.
d.
e.
f.
22, 18, 19, 25, 27, 21, 24
87 21 90 43 54 23 123 110 90 44 50
38, 46, 37, 42, 39, 40, 48, 42
8.4, 7.7, 8.6, 7.5, 8.9, 7.8, 8.6, 9.1, 8.0
515, 720, 635, 895, 585, 690, 770, 840
116, 105, 117, 124, 107, 112, 117, 125, 110, 113