Download Data Mean

Measures of Central Tendency
Copyright © 2009 Pearson Education, Inc.
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Definitions
• An average is a number that is representative
of a group of data.
• The arithmetic mean, or simply the mean is
symbolized by x , when it is a sample of a
population or by the Greek letter mu, , when
it is the entire population.
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Mean
• The mean, is the sum of the data divided by
the number of pieces of data. The formula for
calculating the mean is
Sx
x
n
where Sx represents the sum of all the data and n represents
the number of pieces of data.
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Slide 13 - 3
Example-find the mean
• Find the mean amount of money parents
spent on new school supplies and clothes if 5
parents randomly surveyed replied as follows:
$327 $465 $672 $150 $230
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Solution
327  465  672  150  230 1844
x

 368.8
5
5
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Example
• Malcolm’s average on 5 exams is 81.
Determine the sum of all his exam scores.
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Example
An average of 60 on 7 exams is needed to pass a
course. On her first 6 exams, Sheryl received grades of:
51, 72, 80, 62, 57, and 69.
• What grade must she receive on her last exam to
pass the course?
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Example - continued
• An average of 70 is needed to get a C in the course.
Is it possible for Sheryl to get a C? If so, what grade
must she receive on the 7th exam to get a C in the
course?
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Example - continued
• If her lowest grade of the exams already taken is dropped:
• What grade must she receive on her last exam to pass the
course?
• What grade must she receive on her last exam to get a C
the course?
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Median
• The median is the value in the middle of a set
of ranked data.
• Example: Determine the median of
$327 $465 $672 $150 $230.
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Solution
Rank the data from smallest to largest.
$150 $230 $327 $465 $672
middle value
(median)
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Example: Median (even data)
• Determine the median of the following set of
data: 8, 15, 9, 3, 4, 7, 11, 12, 6, 4.
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Solution
Rank the data:
3 4 4 6 7 8 9 11 12 15
There are 10 pieces of data so the median will
lie halfway between the two middle pieces the 7
and 8.
The median is (7 + 8)/2 = 7.5
3 4 4 6 7 8 9 11 12 15
(median) middle value
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Mode
• The mode is the piece of data that occurs
most frequently.
• Example: Determine the mode of the data set:
3, 4, 4, 6, 7, 8, 9, 11, 12, 15.
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Solution
• The mode is 4 since it occurs twice and the
other values only occur once.
3, 4, 4, 6, 7, 8, 9, 11, 12, 15.
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Midrange
• The midrange is the value halfway between
the lowest (L) and highest (H) values in a set of
data.
lowest value + highest value
Midrange 
2
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Example
• Find the midrange of the data set $327, $465,
$672, $150, $230.
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Example
• Find the midrange of the data set $327, $465,
$672, $150, $230.
150 + 672 822
Midrange 

 411
2
2
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Example
• The weights of eight Labrador retrievers
rounded to the nearest pound are 85, 92, 88,
75, 94, 88, 84, and 101. Determine the
a)
b)
c)
d)
mean
median
mode
midrange
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Example--dog weights 85, 92, 88, 75,
94, 88, 84, 101
a. Mean
85  92  88  75  94  88  84  101 707
x

 88.375
8
8
b. Median-rank the data
75, 84, 85, 88, 88, 92, 94, 101
The median is 88.
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Example--dog weights 85, 92, 88, 75,
94, 88, 84, 101
c. Mode-the number that occurs most
frequently. The mode is 88.
d. Midrange = (L + H)/2
= (75 + 101)/2 = 88
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Measures of Position
• Measures of position are often used to make
comparisons.
• Two measures of position are percentiles and
quartiles.
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To Find the Quartiles of a Set of Data
1. Order the data from smallest to largest.
2. Find the median, or 2nd quartile (Q2), of the
set of data.
3. The first quartile, Q1, is the median of the
lower half of the data; that is, Q1, is the
median of the data less than Q2.
4. The third quartile, Q3, is the median of the
upper half of the data; that is, Q3 is the
median of the data greater than Q2.
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Slide 13 - 23
Example: Quartiles
• The weekly grocery bills for 23 families are as follows.
Determine Q1, Q2, and Q3.
170
330
225
75
95
210
80
225
160
172
270
170
215
130
190
270
240
310
74
280
270
50
81
• Order the data:
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Example: Quartiles continued
• Order the data:
50 75 74 80 81 95 130
160 170 170 172 190 210 215
225 225 240 270 270 270 280
310 330
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Example: Quartiles continued
Q2 is the median of the entire data set which
is 190.
Q1 is the median of the numbers from 50 to
172 which is 95.
Q3 is the median of the numbers from 210 to
330 which is 270.
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Example – Interpreting Statistics
The following statistics represent weekly salaries at the Midtown
Construction Company:
Mean
Median
Mode
$550
$540
$530
1st quartile
$510
3rd quartile
$575
83rd percentile $615
• What is the most common salary?
• What percent of employees’ salaries surpassed $540?
• What percent of employees’ salaries surpassed $575?
• What percent of employees’ salaries were less than $510?
• What percent of employees’ salaries surpassed $615?
• If the company has 100 employees, what is the total weekly
salary of all employees?
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Measures of Dispersion
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Measures of Dispersion
• Measures of dispersion are used to indicate
the spread of the data.
• The range is the difference between the
highest and lowest values; it indicates the
total spread of the data.
Range = highest value – lowest value
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Example: Range
• Nine different employees were selected and
the amount of their salary was recorded. Find
the range of the salaries.
$24,000 $32,000 $26,500
$56,000 $48,000 $27,000
$28,500 $34,500 $56,750
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Solution
• Range = $56,750  $24,000 = $32,750
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Standard Deviation
• The standard deviation measures how much
the data differ from the mean. It is symbolized
with s when it is calculated for a sample, and
with  (Greek letter sigma) when it is
calculated for a population.
s
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
S xx

2
n 1
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To Find the Standard Deviation of a Set
of Data
1. Find the mean of the set of data.
2. Make a chart having three columns:
Data
Data  Mean
(Data  Mean)2
3. List the data vertically under the column
marked Data.
4. Subtract the mean from each piece of data
and place the difference in the Data  Mean
column.
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To Find the Standard Deviation of a Set
of Data continued
5. Square the values obtained in the Data 
Mean column and record these values in the
(Data  Mean)2 column.
6. Determine the sum of the values in the
(Data  Mean)2 column.
7. Divide the sum obtained in step 6 by n  1,
where n is the number of pieces of data.
8. Determine the square root of the number
obtained in step 7. This number is the
standard deviation of the set of data.
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Slide 13 - 34
Example
• Find the standard deviation of the following
prices of selected washing machines:
$280, $217, $665, $684, $939, $299
Find the mean.
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Example
• Find the standard deviation of the following
prices of selected washing machines:
$280, $217, $665, $684, $939, $299
Find the mean.
280  217  665  684  939  299 3084
x

 514
6
6
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Slide 13 - 36
Example continued, mean = 514
Data
217
280
299
665
684
939
Data  Mean
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(Data  Mean)2
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Example continued, mean = 514
Data
217
280
299
665
684
939
Data  Mean
297
234
215
151
170
425
0
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(Data  Mean)2
(297)2 = 88,209
54,756
46,225
22,801
28,900
180,625
421,516
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Example continued, mean = 514
• Find the standard deviation:
s

S xx

2
n 1
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Slide 13 - 39
Example continued, mean = 514
s

S xx

2
n 1
421,516

 84303.2  290.35
5
• The standard deviation is $290.35.
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