![Page 1: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/1.jpg)
Measures of Central
Tendency
![Page 2: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/2.jpg)
Data can be difficult to perceive in raw form
• 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32 3 4 5 56 6 7 8 8 7 65 6 7 8 7 67 6 5 4 43 3 2 2 2 21 1 2 3 4 5 5 4 3 3 4 5 56 6 7 8 9 90 89 78 7 6 5 4 4 3 3 3 4 5 56 6 7 7 8 8 9 8 7 6 65 54 34 4 5 6 76 7 8 98 99 78 6 5 4 3 4 45 55 6 6 7 7 88 9 9 0 0 0 08 6 5 5 4 5 56 6 5 5 6 7 7 8 89 9 9 8 7 7 6 5 5 45 4 5 65 6
![Page 3: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/3.jpg)
A set of numeric data can be described using a single value.
M of CT are values that describe the center of a body of data.
![Page 4: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/4.jpg)
movie
![Page 5: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/5.jpg)
The Mean
Found by dividing the sum of all the data by the number of pieces of data.
x =x
n
Where n is the number of values in the set.
![Page 6: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/6.jpg)
Imagine 3 blocks on a plank.
Where would you place the fulcrum so the blocks would be balanced?
![Page 7: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/7.jpg)
Weighted Mean
Suppose the blocks from the first example all had different weights.
![Page 8: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/8.jpg)
Where x represents each data value, and w represents it’s weight (usually a percentage), or frequency.
x =xw
w
![Page 9: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/9.jpg)
Find the mean for weighted data
EX. Final marks are weighted accordingly:
Tests: 30%
Quizzes: 20%
Assignments: 10%
The student has 78%, 63%, and 12% respectively.
![Page 10: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/10.jpg)
a) Calculate the final mark:
x = 78(0.3) + 63(0.2) + 12(0.1)
(0.3) + (0.2) + (0.1)
= 62
b) What does the student need on the final exam (40%) to get a mark of 70?
![Page 11: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/11.jpg)
62(0.6) + X(0.4) = 70
0.4X = 70 – 37.2
Let X be the exam percent.
37.2 + 0.4X = 70
0.4X = 32.8
X = 82
The student needs 82% on the final exam to raise a final mark of 62% to 70%
![Page 12: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/12.jpg)
A sample of car owners were asked how old they were when they bought their first car.
The results were reported in the following table
Calculate the mean age of the group.
Find the mean for grouped data.
![Page 13: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/13.jpg)
Age Frequency Midpoint (age)
F X M
16-20 10 18 10 X 18 = 180
21-25 18 23 18 X 23 = 414
26-30 12 28 12 X 28 = 336
31-35 8 33 8 X 33 = 264
36-40 2 38 2 X 38 = 76
![Page 14: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/14.jpg)
Age Frequency Midpoint (age)
F X M
16-20 10 18 10 X 18 = 180
21-25 18 23 18 X 23 = 414
26-30 12 28 12 X 28 = 336
31-35 8 33 8 X 33 = 264
36-40 2 38 2 X 38 = 76
![Page 15: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/15.jpg)
Age Frequency Midpoint (age)
F X M
16-20 10 18 10 X 18 = 180
21-25 18 23 18 X 23 = 414
26-30 12 28 12 X 28 = 336
31-35 8 33 8 X 33 = 264
36-40 2 38 2 X 38 = 76
![Page 16: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/16.jpg)
To calculate the mean
X = 180 + 414 + 336 + 264 + 76
x =(f X m)
f
10 + 18 + 12 + 8 + 2
X = 25.4
The mean age is 25.4
![Page 17: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/17.jpg)
Median: The middle value of an ordered set
Note: If the set has an even number of data points, find the mean of the two middle-most values.
![Page 18: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/18.jpg)
Find the median:
4, 2, 8, 5, 9, 1, 4, 6, 8, 2, 9
Re-order:
1, 2, 2, 4, 4, 5, 6, 8, 8, 9, 9
Identify the median: 5
![Page 19: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/19.jpg)
Mode: The most frequent value
Find the mode:
1 3 5 5 7 8 9 11 14 17
The mode is 5
![Page 20: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/20.jpg)
Outliers
An element of the data set that is very different from the others.
If there is sufficient reason, this element may be ignored.
![Page 21: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/21.jpg)
An example of an outlierA student wanted to determine the most
favorable temperature that people wanted to experience on their holidays.
She took a survey and asked the following question:
What temperature do you prefer when you are looking to travel on the holiday?
Her results were as follows:
![Page 22: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/22.jpg)
All temperatures are in OC
32 31 29 26 4044 33
34 -20 39 41 2829 35
![Page 23: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/23.jpg)
What measure should you use?
If data contains outliers, use the median.
If the data are roughly symmetric, use the mean or the median.
If the data are qualitative (eg election), use the mode
![Page 24: Measures of Central Tendency Data can be difficult to perceive in raw form 4 5 56 6 4 23 4 5 6 76 7 56 5 54 4 3 5 6 7 8 9 9 00 9 8 78 6 6 5 4 4 3 32](https://reader035.vdocuments.site/reader035/viewer/2022081519/56649ec05503460f94bcbca5/html5/thumbnails/24.jpg)
Page 1821a, 2, 5