Slide Copyright © 2007 Pearson Education, Inc Publishi.docx

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Copyright © 2007 Pearson Education, Inc Publishing as Pearson
Addison-Wesley.
Measures of Center
Addison-Wesley.
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Key Concept
When describing, exploring, and comparing data sets, these
characteristics are usually extremely important: center,
variation, distribution, outliers, and changes over time.
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What are we going to learn? Measures of Center
Mean, Median, Mode Range Standard Deviation
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Arithmetic Mean
(Mean)
You hear people sometimes say average
the measure of center obtained by adding the values and
dividing the total by the number of values
Definition
N
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Notation
denotes the sum of a set of values (sigma)
x is the variable usually used to represent the individual
data values.
n represents the number of values in a sample (because it is
lower case)
N represents the number of values in a population (because it
is upper case)

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Notation
µ is pronounced ‘mew’ and denotes the mean of all values
in a population
is pronounced ‘x-bar’ and denotes the mean of a set of sample
values
Symbol for the sample mean
Symbol for the population mean
This equation reads the mean equals the sum of each score
divided by n
x =
n
x
N
µ =
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Let’s Practice Some Mean Exercises
Get out a calculator
Determine the mean for each of the data sets:

3, 5, 7
6, 5, 6, 2, 1
3, 21, 16, 9, 13
7, 3, 6, 3, 2
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Let’s Practice Some Mean Exercises
Get out a calculator
Determine the mean for each of the data sets:
3, 5, 7 = 3 + 5+ 7 = 15/3 =
b) 6, 5, 6, 2, 1 = 6 + 5 + 6 + 2 + 1 = 20/5
3, 21, 16, 9, 13 = 3 + 21 + 16 + 9 + 13 = 62/5
7, 3, 6, 3, 2 = 21/5
Addison-Wesley.
Answers: a) 5; b) 4 c) 12.4 ;d) 4.2
Don’t assume you know this. Please practice with your
calculator.
Addison-Wesley.
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Let’s Try this Exercise

The table shows daily sales in the first 2 weeks of July in
Windsor Mill at the sandwhich shop.
DateSalesJuly 146July 245July 396July 426July 585July
645July 765July 884July 964July 1044July 1185July 1225July
1345July 1485
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What is the mean sales for the data?
56.7
59
60
63.5

1345July 1485
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Answer
What is the mean sales for the data?
56.7
59
60
63.5
1345July 1485

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Definitions
Mode
the value that occurs most frequently
Mode is not always unique
A data set may be:
Bimodal (two modes)
Multimodal (many modes)
No Mode
Mode is the only measure of central tendency that can be used
with nominal data

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a. 5.40 1.10 0.42 0.73 0.48 1.10
b. 27 27 27 55 55 55 88 88 99
c. 1 2 3 6 7 8 9 10
Mode - Examples
Mode is 1.10
Bimodal - 27 & 55
No Mode
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Let’s Practice Finding the Mode
Find the mode in the following data sets:
17, 19, 19, 4, 19, 26, 3, 21, and 19
3, 4, 4, 5, 2, 1, 3, 0, 0, 3
0, 0, 2, 1, 1 5, 6, 3, 7, 0, 3, 1
Answers: a) 19; b) 3 c) 0 and 1
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Definitions
Median
the middle score is not affected by an extreme value
often denoted by x (pronounced ‘x-tilde’)

~
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Finding the Median Steps
Step 1 – Place the data in an ordered array (from lowest to
highest)
Step 2 – Determine N
Step 3a - If the number of scores (N) is odd, the median is the
number located in the exact middle of the list.
Step 3b - If the number of scores (N) is even, the median is
found by computing the mean of the two middle numbers.
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5.40 1.10 0.42 0.73 0.48 1.10 0.66
0.42 0.48 0.66 0.73 1.10 1.10 5.40
(in order - odd number of values)
exact middle MEDIAN is 0.73
5.40 1.10 0.42 0.73 0.48 1.10
0.42 0.48 0.73 1.10 1.10 5.40
0.73 + 1.10
2
(in order - even number of values – no exact middle
shared by two numbers)
MEDIAN is 0.915
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Let’s Practice Finding the Median
Find the median for the following data set:
A) 92, 2, 90, 86, 96B) 12.5, 10.5, 24.5, 26, 15, 21C) 250, 240,
188, 198, 240, 230, 210, 225
Answers: A) 90; B) 18; C) 227.5
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1345July 1485
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What is the median for the data?
50
55
58
60DateSalesJuly 146July 245July 396July 426July 585July

1345July 1485
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Answer
What is the median for the data?
50
55
58
60DateSalesJuly 146July 245July 396July 426July 585July
1345July 1485

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Addison-Wesley.
Q1, Q2, Q3
divide ranked scores into four equal parts
Quartiles
25%
25%
25%
25%
Q3

Q2
Q1
(minimum)
(maximum)
(median)
Addison-Wesley.
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Addison-Wesley.
Definition
Q1 (First Quartile) separates the bottom 25% of
sorted values from the top 75%. Q2 (Second Quartile) same as
the median; separates the bottom 50% of sorted values
from the top 50%. Q1 (Third Quartile) separates the bottom
75% of sorted values from the top 25%.
Addison-Wesley.
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Here are the steps…Step 1 – Put the data into an ordered array
(from lowest to highest)Step 2 – Find the median for the entire
data set (Q2)Step 3 – Find the median for the first half of the
data set (Q1)Step 4 – Find the median for the 2nd half of the
data set (Q3)

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Step 1
0, 11, 22, 2, 3, 6, 21, 32, 2, 2, 9, 17, 21, 14, 19
0, 2, 2, 2, 3, 6, 9, 11, 14, 17, 19, 21, 21, 22, 32
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Step 2
0, 2, 2, 2, 3, 6, 9, 11, 14, 17, 19, 21, 21, 22, 32
Q2
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To find the top/bottom half, put a line through Q2
0, 2, 2, 2, 3, 6, 9, 11, 14, 17, 19, 21, 21, 22, 32
The top half of the data becomes 0, 2, 2, 2, 3, 6, 9.
The bottom half of the data becomes 14, 17, 19, 21, 21, 22, 32
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Step 3
0, 2, 2, 2, 3, 6, 9
Q1
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Step 4
14, 17, 19, 21, 21, 22, 32
Q3
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So the 3 data points are:
Q1 = 2; Q2 = 11; Q3 = 21
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Addison-Wesley.
Interquartile Range (or IQR): Q3 - Q1
Some Other Statistics Five Number Summary:
Min, Q1, Q2, Q3, Max
2
Q3 - Q1 Semi-interquartile Range:
2
Q3 + Q1
Midquartile:
Addison-Wesley.
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1345July 1485
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What is the five-number summary for the data?
a) 26, 46, 55, 86, 96
b) 25, 45, 55, 85, 96
c) 26, 46, 56, 85, 96
d) 25, 46, 55, 86, 96
1345July 1485

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Answer
What is the five-number summary for the data?
a) 26, 46, 55, 86, 96
b) 25, 45, 55, 85, 96
c) 26, 46, 56, 85, 96
d) 25, 46, 55, 86, 96
1345July 1485

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Midrange
the value midway between the maximum and minimum
values in the original data set
Definition
Midrange =
maximum value + minimum value
2

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Carry one more decimal place than is present in the original set
of values.
Round-off Rule for
Measures of Center
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Best Measure of Center
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Skewness
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Recap
In this section we have discussed: Types of measures of center
Mean
Median
Mode Best measures of center Skewness

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Addison-Wesley.
Measures of Variation
Addison-Wesley.
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Key Concept
Because this section introduces the concept of variation, which
is something so important in statistics, this is one of the most
important sections in the entire book.
Place a high priority on how to interpret values of standard
deviation.
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Let’s try some basic computations

Addison-Wesley.
Determine Σxy and Σx2x2247y6156
Addison-Wesley.
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Addison-Wesley.
Determine Σxy
Determine the sum of x times y
12 + 2 + 20 +42 = 76x2247y61561222042

Addison-Wesley.
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Addison-Wesley.
Determine Σx2
Determine the sum of x squared
4 + 2 + 16 + 49 = 71x2247x2421649
Addison-Wesley.
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Now try one on your own
Addison-Wesley.
Determine Σxy and Σx2x3239y5356

Addison-Wesley.
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Did you get?
Addison-Wesley.
Determine Σxy and Σx2
90 and 103x3239y5356
Addison-Wesley.
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Definition
The range of a set of data is the difference between the
maximum value and the minimum value.
Range = (maximum value) – (minimum value)

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Definition
The standard deviation of a set of sample values is a measure of
variation of values about the mean.
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Sample Standard
Deviation Formula
Let’s read this formula aloud. S which stands for standard
deviation is equal to the square root of the sum of each score
minus the mean squared divided by n-1.
- x)2
n - 1
s =
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Standard Deviation -
Important Properties The standard deviation is a measure of
variation of all values from the mean. The value of the
standard deviation s is usually positive. The value of the
standard deviation s can increase dramatically with the
inclusion of one or more outliers (extreme scores). The units
of the standard deviation s are the same as
the units of the original data values.
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Population Standard
Deviation
2
- µ)
N
This formula is similar to the previous formula, but instead, the
population mean and population size are used.

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Definition The variance of a set of values is a measure of
variation equal to the square of the standard deviation.
Sample variance: Square of the sample standard deviation s
Population variance: Square of the population
standard deviation
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Variance - Notation
standard deviation squared
s
2
2
}
Notation
Sample variance
Population variance
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Round-off Rule
for Measures of Variation

Carry one more decimal place than is present in the original
set of data.
Round only the final answer, not values in the middle of a
calculation.
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Let me now show you how to computer s (standard deviation)I
do it very simply…with a chartLet’s consider that we want to
find the standard deviation for the following data set: 5, 4, 2, 3,
6x (each score)x – mean (x-mean) squared
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Let me now show you how to computer s (standard
deviation)Let’s consider that we want to find the standard
deviation for the following data set: 5, 4, 2, 3, 6Step 1 – Place
each score in the chart and sum.Step 2 – Find the mean.Step 3 –
Identify N

Total this rowX (each score)x – mean (x-mean) squared
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deviation for the following data set: 5, 4, 2, 3, 6Step 1 – Place
each score in the chart and sum.
Total this rowX (each score)x – mean (x-mean) squared5423620

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deviation for the following data set: 5, 4, 2, 3, 6Step 2 – Find
the mean
Total this row
20/5 = 4
Mean = 4X (each score)x – mean (x-mean) squared5423620
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deviation for the following data set: 5, 4, 2, 3, 6Step 3 –
Identify N
Total this row
Mean =4
N = 5X (each score)x – mean (x-mean) squared5423620

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deviation for the following data set: 5, 4, 2, 3, 6Step 4 – Now
subtract each score from the mean
Total this row
Mean =4
N = 5X (each score)x – mean (x-mean) squared55-4 = 144-4=
022-4= -233-4= -166-4= 220

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deviation for the following data set: 5, 4, 2, 3, 6Step 5 – Total
this column. The total should almost always be 0.
Total this row
Mean =4
N = 5X (each score)x – mean (x-mean) squared51402-23-
162200
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deviation for the following data set: 5, 4, 2, 3, 6Step 6 – Square
each score.
Total this row
Mean =4

N = 5
Squared means the number times itself. On your calculator is
an x squared button.
Do not put negative numbers in your calculator. The last
column will always be positive.X (each score)x – mean (x-
mean) squared511 x 1 = 1400 x 0 = 02-2-2 x -2 = 43-1-1 x -1 =
1622 x 2 = 42000 x 0 = 0
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deviation for the following data set: 5, 4, 2, 3, 6Step 7 – Total
this column.
Total this row
Mean =4
N = 5
Squared means the number times itself. On your calculator is
an
This total becomes the numerator in the formula.X (each score)x
– mean (x-mean) squared5114002-243-1162420010

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Sample Standard
Deviation Formula
10
Now the formula comes to life and becomes…
Original Formula
n
- x)2
n - 1
s =
5 - 1

s =
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Sample Standard
Deviation Formula
10
10
Watch how the formula works itself out…
5 - 1
s =
4
s =

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Sample Standard
Deviation Formula
10
2.5
Watch how the formula works itself out…
4
s =
s =
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Sample Standard
Deviation Formula
2.5
Compute the square root of 2.5

s = 1.58
Voila!
s =
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Sample Standard Deviation FormulaI know that was a lot to take
in at once; however, let’s try one now by yourself. Just take
your time. Standard deviation takes practice to get perfect, but
with a little practice you will get it.
Take your time and check your work. DON’T RUSH!
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Sample Standard Deviation FormulaFind s for the following
data sets:A) 4, 2, 3B) 9, 6, 7, 2C) 2, 6, 1D) 498, 500, 503, 498,
501
Answers: a) 1; b) 2.94; c) 2.646; d) 2.12
Do not move forward unless you have the answers.
I mean it!
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Definition
Empirical (68-95-99.7) Rule

For data sets having a distribution that is approximately bell
shaped, the following properties apply: About 68% of all values
fall within 1 standard deviation of the mean. About 95% of
all values fall within 2 standard deviations of the mean.
About 99.7% of all values fall within 3 standard deviations of
the mean.
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The Empirical Rule
Slide *
The Empirical Rule
Slide *
The Empirical Rule
Slide *

What does standard deviation mean?Standard deviation is a
measure which talks about how the scores scatter around the
mean.If s is low, the scores, in general, are closer to the mean.If
s is high, the scores, in general, are farther from the mean.
Think of yourself as the mean and you are holding jellybeans in
your clinched fist. You open your hand and drop them at your
feet.Some jellybeans will be right by your feet and a few may
scatter a few feet away. S would be low. In general, the
jellybeans are staying close to you.
Think of yourself again as the mean and you are holding
jellybeans in your clinched fist. This time, you open your hand
and toss the jellybeans in front of you. S will be high. In
general, some jellybeans are near you, but more of them are
away from you.
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What else does standard deviation mean?There are 3 steps on
the standard deviation: 68%, 95%, and 99%
How many steps? Three
And what are they? 68% 95% 99%
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Here is standard deviation at workImagine the first statistics
exam. The mean is 80 and the standard deviation for our class
is 5.
Slide *
The Empirical Rule
80

mean
Slide *
The Empirical Rule
80 + 5 = 85
s = 5
85
75
80 - 5 = 75
Finding: 68% of the class scored between 75 and 85 on the first
STAT exam.
Slide *
The Empirical Rule
75 - 5 = 70
85 + 5 = 90
70
90
STAT exam.
Slide *

70 - 5 = 65
90 + 5 = 95
65
95
STAT exam.
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Definition
An outlier is a value that is located very far away from
almost all of the other values.
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Important Principles
An outlier can have a dramatic effect on the mean. An outlier
can have a dramatic effect on the standard deviation. An
outlier can have a dramatic effect on the scale of the histogram
so that the true nature of the distribution is totally
obscured.
Slide *

Addison-Wesley.
Key Concept
This section introduces measures that can be used to compare
values from different data sets, or to compare values within the
same data set. The most important of these is the concept of the
z score.
Addison-Wesley.
Slide *
Addison-Wesley.
z Score (or standardized score)
the number of standard deviations that a given
value x is above or below the mean
Definition
Addison-Wesley.
Slide *
Addison-Wesley.
Sample

Round z to 2 decimal places
Measures of Position z score
z =
Population
x - µ
z =
x - x
s
Addison-Wesley.
Slide *
Addison-Wesley.
Interpreting Z Scores
Whenever a value is less than the mean, its corresponding z
score is negative
Ordinary values: z score between –2 and 2 Unusual
Values: z score < -2 or z score > 2
Addison-Wesley.

Slide *
Let’s compute some z-scoresA normal distribution has a mean
of 50 and a standard deviation of 8.Compute the z-score forA)
X = 56
B) X = 43
C) X = 68
Answers: a) 0.75; b) -0.875; c) 2.25
56-50
8
43-50
8
68-50
8
Slide *
Addison-Wesley.
For a set of data, the 5-number summary consists of the
minimum value; the first quartile Q1; the median (or second
quartile Q2); the third quartile, Q3; and the maximum value. A
boxplot ( or box-and-whisker-diagram) is a graph of a data set
that consists of a line extending from the minimum value to the
maximum value, and a box with lines drawn at the first quartile,
Q1; the median; and the third quartile, Q3.
Definitions
Addison-Wesley.

Slide *
Addison-Wesley.
Boxplots
A boxplot is another way to display data.
Addison-Wesley.
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Addison-Wesley.
Boxplots - cont
Addison-Wesley.
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Addison-Wesley.
Boxplots - cont
Addison-Wesley.

Slide *
Addison-Wesley.
Boxplots - cont
Is this distribution left or right skewed?
Right-skewed
Addison-Wesley.
Slide *
Practice Exercise for Chapter 3
The amounts of fuel oil (in gallons) consumed for a sample of
both newer and older homes is reported below:
Newer homes: 600, 590, 605, 600, 603, 610, 597, 595
Older homes: 810, 750, 790, 800, 850
Determine the standard deviation and variance for newer homes
Determine the standard deviation and variance for older homes
Which homes are using less oil and why? (hint: use the 3 steps
rule)
Compute the z score for the new home that consumes 597
gallons.
Compute the z score for the old home that consumes 790
gallons.
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Send me your work directly at [email protected] for the Practice
Exercise answers.

Addison-Wesley.
Addison-Wesley.
s
Congratulations. You have just been hired as the new CEO for
Handback Industries. You were excited until you started and
within first the few days, the director of HR came into your
office and indicated that the employees were threatening to
strike and go to the media if things were not fixed immediately.
Since you were successful in your Statistics course taken during
the Summer of 2013, you decide that you will assist the HR
Director with her analysis.
Problem 1 - The employees have indicated that 95% of
employees in one of the departments are receiving higher
salaries than any other department because their supervisor
parties with them each weekend. Some employees only have an
hourly rate which you have to calculate their yearly salary
before completing the calculations. Please calculate the 95%
standard deviation for each department and indicate for me if
any department is receiving on average $2k or more on the high
end.
Department 1
Department 2
Department 3
Department 4
Department 5
$22,450
18,550
B
22555
19050
$30,050

27,050
16765
29950
21545
$23,785
26,785
30050
23785
27950
A
25050
29955
B
27550
$27,594
26557
A
27650
25550
A – $9/hr @ 40 hrs week @ 50 weeks a year
B – $11/hr @ 35 hrs wee @ 52 weeks a year
Is any department receiving on average $2k or more on the high
end of the standard deviation using standard deviation at 95%?
If so, which department and be sure to explain your answer.
Problem 2
Find the five number summary for the data for all five
departments. Consider all five departments one department.
Problem 3
Find the Q3 and median for Departments 1 and 3 individually.

Which department has a higher Q3? Which department has a
higher median?
Problem 4
Find the Q1 and median for Departments 2 and 4 individually.
Which department has a higher Q1? Which department has a
higher median?

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