1. Measures of Central Tendency “Where is the Middle?”
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Measures of Central Tendency “Where is the Middle?”
To accompany Hawkes Lesson 3.1
Original content by D.R.S.
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2. The Mean The street name for “mean” is “average”.
But be professional and call it “the mean”
In words: “Add up all the numbers. Divide by how many numbers there are in the list.
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3. Formula for calculating the Mean
In symbols: 𝜇= 𝑥 𝑖 𝑁 or 𝑥 = 𝑥 𝑖 𝑛
𝑥 𝑖 are the individual data values
There are 𝑁 data values in the population.
Or if a sample, little 𝑛 data values.
𝑖 goes from 1, 2, 3, …, 𝑁 or 𝑛
Greek letter Σ, capital sigma, means “sum”
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4. Formula for calculating the Mean
In symbols: 𝜇= 𝑥 𝑖 𝑁 or 𝑥 = 𝑥 𝑖 𝑛
Rounding: one more decimal place than the data’s precision
The mean of { 15, 15, 17 } is 15.7
The mean of { 15.0, 15.0, 17.0} is 15.67
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5. Which name to use? 𝜇 or 𝑥 ? 𝜇 is the Greek letter mu, pronounced “mew”
Use 𝜇 if your calculation is the mean of the entire population being studied.
𝑥 is pronounced “bar x”.
Use 𝑥 if your calculation is the mean of just a sample of the entire population.
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6. Example – The Mean
Data: Fernwood home sales at 3 common data.pptx Slide #2
The mean selling price of homes in Fernwood in the given month is $__________.
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7. The Median Arrange the values in order from lowest to highest.
Take the “middle” number. That’s the median.
Example: We rolled five dice and recorded the totals. Our results: 10, 12, 17, 18, 18, 21, 23
Count: n = 7, an odd number of numbers.
Who’s in the middle?
The median is _____.
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8. The Median of an even # of #s
Same experiment but this time we got these results: 10, 12, 17, 17, 18, 19, 20, 22
Count: n = 8, an even number of numbers.
There are two middle numbers!
10, 12, 17, 17, 18, 19, 20, 22
In this case, take the midpoint of the two middle numbers: ( ) ÷ 2 = ______.
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9. Example – The Median Fernwood example: 3 common data.pptx Slide #2
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10. The Mode What value occurs most frequently?
Rolling one die over and over and recording the results.
Data: 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6
The mode is _____.
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11. The Modes, plural What value occurs most frequently?
Rolling one die over and over and recording the results.
Data: 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6
The modes are _____ and _____.
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12. The Mode – another special case
What value occurs most frequently?
Prices of ten cars at Miracle Motors:
$3, $4, $8, $9,100 $9,399
$10,599 $12,399 $15,999 $17,999 $32,999
Everybody is tied with one occurrence.
“There is No Mode.”
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13. More about No Mode What value occurs most frequently?
Record the color of each traffic signal we encounter on our travels
R, R, G, G, G, R
Red and Green are tied with three occurrences each. “There is No Mode.”
But if R, R, G, Y, G, G, R, then we have two modes, Red and Green.
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14. How many modes? 1, 2, many, or 0
How many modes does the data set have?
We say that the data set is ___________.
There is 1 mode.
UNIMODAL
There are 2 modes.
BIMODAL
There are >2 modes.
MULTIMODAL
There is no mode.
“There is no mode”
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15. Fernwood, continued Data: 3 common data.pptx Slide #2
The mean home price is $___________.
The median home price is $__________.
The mode of the home price is $_________.
The range of home prices is the highest minus the lowest = $________.
The midrange is 𝑙𝑜𝑤𝑒𝑠𝑡+ℎ𝑖𝑔ℎ𝑒𝑠𝑡 2 = $_________, halfway between the extreme values.
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16. “a Statistic” vs. “a Parameter”
If the mean or median or mode or anything else was calculated from an entire population’s data – then it’s called a parameter
If the mean or median or mode or anything else was calculated from just a sample – then it’s called a statistic
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17. The lingo of Samples, Populations
a SAMPLE
a POPULATION
We calculate STATISTICS
There are 𝑛 items
The mean is denoted as 𝑥
We calculate PARAMETERS
There are 𝑁 items
The mean is denoted as 𝜇
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18. Mean for a Frequency Distribution
Previously: a list of numbers
Now: a list of numbers and counts of how many times each number occurs
Same formula but with some adaptations
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19. Example of a Frequency Distribution
Bowling League – bowlers’ averages, Slide #3 in 3 common data.pptx or Excel in 3 common data.xlsx
There are “classes”, ranges of scores.
There are counts of how many individuals belong to that class.
What about the boundaries?
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20. Example – The Mean of a Frequency Distribution
What is the mean league average?
𝑥 𝑖 𝑛 becomes 𝑓 𝑖 ∙ 𝑥 𝑖 𝑓 𝑖
The data values are denoted as 𝑥 𝑖
For 𝑖=1, 2, 3,…
Value 𝑥 𝑖 occurs 𝑓 𝑖 times
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21. Example – The Mean of a Frequency Distribution
What is the mean league average?
𝑥 𝑖 𝑛 becomes 𝑓 𝑖 ∙ 𝑥 𝑖 𝑓 𝑖
The numerator contains a shortcut for repeatedly adding the same number, such as a 150 score * how many individuals.
The denominator adds up all the frequencies to get the total how many items in the set.
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22. Example – Mean for Frequency Dist.
PPT – Slide #4 in 3 common data.pptx or Excel “Bowling mean” sheet in the workbook 3 common data.xlsx
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23. Example – Mode for a Frequency Distribution
Frequency means “How many times did the value occur?”
So it’s easy – just look for the class which has the highest frequency.
The mode is the VALUE associated with that highest frequency, not the count itself.
You could say “the modal class is 90 and below”
Or you could say “the mode is 85”.
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24. Example – Mode for a frequency distribution
Refer again to the bowling league: Slide #3 at 3 common data.xlsx
“The Modal Class is ________________” (not a number, but a class name).
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25. The Weighted Mean The classic example is GPA, Grade Point Average.
The data values are A=4, B=3, C=2, D=1, F=0
The weights are the courses’ credit values
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26. Weighted Mean Example Original data: Slide #5 in 3 common data.pptx
Computation: Slide #6 in 3 common data.pptx
Excel spreadsheet for experimentation: Sheet “GPA experiment” in workbook 3 common data.xlsx
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27. Weighted Mean Example What effect did the 4-credit course have?
Notice the convention for GPA is two decimal digits; usually it would be one more digit than the data, or 3.1.
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28. TI-84 Put the data values into a TI-84 LIST. STAT menu, CALC submenu
1-Var Stats Ln (if omitted, it uses L1)
𝒙 , n, Med are among the values we get
minX and maxX help you find range, midrange
Mode must still be done by hand
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29. TI-84 Fernwood example
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30. Excel Fernwood example*
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31. Which measure “best” describes the data?
Consider the Fernwood data again. Which one number “best” communicates the housing market?
Mean?
Median?
Mode?
Midrange?
Experiment with the Excel file to explore more
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32. Which measure “best” describes the “middle” of the data?
If the data is qualitative, use the Mode.
MODE
Have to use Mode for categorical, non-numerical data!
Can’t take the “average” of non-numerical, non-rankable data.
If the data is quantitative, Mean is Nice.
MEAN
But if there are OUTLIERS, they can distort the mean.
Or if the data is greatly SKEWED, the mean might be distorted.
Sometimes Median is better
MEDIAN
The Median is more resistant to distortion from outliers
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Credit for idea: Hawkes Learning Systems online instruction page

33. Trimmed Mean Sometimes used in cases like Fernwood home prices.
We throw out some of the extreme values on either end.
Similar effect to median – to insulate the result from extreme values on the ends.
But the satisfaction of having a mean that lets all the middle numbers have their say.
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34. Trimmed Mean Example “the 20% trimmed mean”
We have 15 values in the data set
20% of 15 = 3: Ignore 3 lowest, 3 highest
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35. TI-84 Frequency Distribution
You have to use TWO lists:
One list for the data values
Another list for the frequencies
Then 1-Var Stats L1,L2
Tell it the data value list first, frequency list second
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36. Frequency Distribution TI-84 for the bowling league
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37. Weighted Mean Each value has a “weight”
Because some values are “more important” than others.
Those values are “weighted more heavily”
And other values are “less important”
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38. Your GPA A classic Weighted Mean situation
Course
Grade (Points)
Credits
British Literature I
B (3) 3
U.S. History
A (4)
Physics 4
French II
C (2)
Ice Fishing 1
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39. TI-84 Weighted Mean You have to use TWO lists:
First list for the data values (the grades’ values)
Second list for the weights (how many credits)
Then 1-Var Stats L1,L2
Careful! Easy to mix these up, especially with GPA problems, since both lists are values in the same 0-4 range. (Example: “A” in 1-credit P.E. would be treated as a “D’ in a 4-credit course!)
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40. TI-84 GPA Example
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