Download central tendency

Central Tendency
Objectives:
…to calculate measures of central tendency and to decide which measure is
most appropriate for a given situation.
…to construct, compare, and identify parts of a box-and-whisker plot
…to construct, compare, and identify parts of a stem-and-leaf plot
Assessment Anchor:
7.E.2.1 – Describe, compare, and/or contrast data using measures of
mean, median, mode or range.
Vocabulary alert!!
CENTRAL TENDENCY – an attempt to find the “average”
or “central value” of a given set
of data
MEAN – the sum of the data items divided by the number of
data items
MEDIAN – the middle data item found after sorting the data
items in ascending order (could be the mean of
two middle numbers if the data set has an even
number of items)
MODE – the data item that occurs most often
RANGE – the difference between the highest and the lowest
data item
OUTLIER – a data item that is much higher or much lower
than all the other data items
Central Tendency
NOTES
In most cases, the data set is a list of numbers. The “average” is a single
number used to identify the typical value of the data set. While the “mean” is the
most common measure to use, there may be situations where either the “median”
or the “mode” is a more appropriate measure of the central tendency of the data.
While we want to know HOW to calculate these measures of central
tendency, we would like to also focus on WHY different situations call for the use
of different measures of central tendency. Sometimes the determining factor lies in
the kind of data, other times it has to do with the “spread” of the data, and other
times it simply comes down to what kind of argument one wishes to make using
the data.
EXAMPLES
The following table shows the point totals for the Exeter boys’ basketball team:
Game
Points
1
54
2
42
3
42
4
53
5
98
6
48
7
49
8
34
9
51
10
48
11
42
MEAN: 51
<-------- add the numbers to get a total of 561, then divide by 11
MEDIAN: 48
<-------- order the numbers, then find the middle number
34, 42, 42, 42, 48, 48, 49, 51, 53, 54, 98
MODE: 42
<-------- find the number that happens most often
RANGE: 64
<-------- subtract the smallest number from the largest number
OUTLIER: 98
<-------- this number is MUCH higher than the rest!
“So… what is the average number of points scored
by the Exeter boys’ basketball team?
“What if the outlier was removed?”
Central Tendency
The following table shows the temperatures (in ºF) for the last 10 days in March:
Tues.
45º
Wed.
38º
Thurs.
41º
Fri.
48º
Sat.
46º
Sun.
48º
Mon.
45º
Tues.
39º
Wed.
36º
Thurs.
40º
MEAN: 42.6
<-------- add the numbers to get a total of 426, then divide by 10
MEDIAN: 43
<-------- order the numbers from smallest to largest, find the
middle two numbers, add them up and divide by 2
36, 38, 39, 40, 41, 45, 45, 46, 48, 48
MODE: 45 and 48
<-------- find the numbers that happen the most!
RANGE: 12
<-------- subtract the smallest from the largest number!
OUTLIER: none
<-------- there are no numbers MUCH higher or MUCH
lower than the rest!
The following table shows the test scores for students in Mr. Seidel’s period 9
class:
Student
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Test Score
44
79
81
88
79
85
86
90
90
77
85
90
84
78
MEAN:
MEDIAN:
MODE:
RANGE:
OUTLIER:
Central Tendency
STEM-AND-LEAF PLOTS
***A stem-and-leaf plot is ONE way to organize data by showing each item in
order from smallest to largest. The leaf is the last digit to the right… the stem is
the remaining digit or digits.
39
3
stem
24.2
9
24
leaf
stem
2
leaf
To make a stem and leaf plot:
GIVEN DATA:
1. List the stems and draw a line
to the right of the stems.
(keep stems sequential)
2. Arrange the leaves on each stem
from smallest to largest.
(each data item gets a leaf)
3. Include a KEY to explain how
to read the plot.
38, 49, 38, 41, 50, 33, 45, 36, 53
STEM-AND-LEAF PLOT:
3 3688
4 159
5 03
Key: 4 | 5 means 45
Make a stem-and-leaf plot
HR per season by Terra Coveroff:
Interest rates given by 6 local banks:
46, 49, 30, 61, 39, 38, 34, 40, 33, 38
0.4%, 1.1%, 1.3%, 0.7%, 0.2%, 0.3%
Central Tendency
Read a stem-and-leaf plot
8
9
10
11
47
788
025559
2
1. What is the median?
2. What is the range?
3. What is the mode?
Key: 9 | 7 means 9.7
27 2 4 5 5 5
28 9 9
29 0 0 2 4 5 7
1. What is the price range?
2. How many prices are $2.90 or higher?
Key: 25 | 8 means $2.58
3. What is the median price?
BOX-AND-WHISKER PLOTS
***A box-and-whisker plot is ANOTHER way to organize data. The data is
divided into four equal parts called quartiles. The median separates the data into
two halves. The lower quartile is the median of the lower half…the upper quartile
is the median of the upper half.
To make a box-and-whisker plot:
GIVEN DATA:
1. Find the 5 SPECIAL POINTS…
the lowest, the highest, the median,
the lower quartile, the upper quartile
2. Draw a number line that can hold
all the data.
52, 62, 90, 50, 58, 61, 35, 48, 60
35, 48, 50, 52, 58, 60, 61, 62, 90
lowest
3. Plot the special points and create
the picture.
•
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49
median
lower quartile
•
61.5
highest
upper quartile
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35
40
45
50
55
60
65
70
75
80
85
90
Central Tendency
Make a box-and-whisker plot
Make a box-and-whisker plot for the following numbers:
78, 90, 93, 100, 76, 88, 80, 90, 52, 78, 80, 94
Order them:
5 special points:
Number line and plot:
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45
50
55
60
65
70
75
80
85
90
95
100
Read a box-and-whisker plot
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85
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90
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95
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100
105
110
1. What is the median?
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2. What is the lower quartile?
3. What is the range?
4. How many data items are there?
The box-and-whisker plot below shows the prices of CDs in a store:
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6
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8
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10
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12
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14
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16
1. What is the lowest priced CD?
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2. What is the range of the costs?
3. What is the median cost?
4. How many CDs are under $10?