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Warm Up
Find the variance for the given data. Round your answer to one
more decimal place than the original data.
77251
Find the standard deviation for the given sample data. Round
your answer to one more decimal place than is present
inthe original data.
18 18 18 9 15 5 10 5 15
RECALL:Sample Standard
Deviation Formula
( x  x )
s
n 1
x bar is the sample mean
n is the number or sample values
Sigma means add
s is the standard deviation
2
RECALL: Variance is the square of standard
deviation.
s
( x  x )
n 1
2
( x  x )
s
n 1
2
To find variance
1.Find the mean=xbar
2. Find n and n-1
3. Find each x-xbar (subtract
xbar from each data
value.)
4. Square each x-xbar
5. Add these and divide by n1
To find standard deviation
1. Find the mean=xbar
2. Find n and n-1
3. Find each x-xbar (subtract
xbar from each data
value.)
4. Square each x-xbar
5. Add these and divide by n1
6. Square root
Solution#1
Find the variance for the given data. Round your
answer to one more decimal place than the
original data.
77251
To find the variance:
Find the mean = 4.4
Find n = 5 so n-1 = 4
Find each x-xbar = 2.6, 2.6,-2.4, .6, -3
Square each x-xbar = 6.76,6.76,5.76,.36,11.56
Add these and divide by n-1 = 7.8
The variance is 7.8.
Solution#2
Find the standard deviation for the given sample data. Round
your answer to one more decimal place than is present in
the original data.
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To find the standard deviation:
Find the mean = 12.56
Find n = 9 so n-1 = 8
Find each x-xbar = 5.44, 5.44,5.44, -3.56, 2.44,-7.56, -2.56,
-7.56, 2.44
Square each x-xbar =29.64, 29.64, 29.64, 12.64, 5.98,
57.09, 6.53, 57.09, 5.98
Add these and divide by n-1 = 29.29
6. Square root = 5.41 (Note: we do not square root for
variance.)
The standard deviation is 5.41
Chapter 3
Statistics for Describing, Exploring, and
Comparing Data
3-1 Review and Preview
3-2 Measures of Center
3-3 Measures of Variation
3-4 Measures of Relative Standing and
Boxplots
FIRST-Empirical (or 68-95-99.7) Ru
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.
The Empirical Rule
The Empirical Rule (In English)
Assuming the values are normally distributed,
Most (95%) values are within 2 standard deviations of
the mean.
A value within 2 sd’s of the mean is considered USUAL
A value outside of 2 sd’s of the mean is considered
UNUSUAL
Example
The author of the text measured his pulse rate to be
48 beats per minute.
Is that pulse rate unusual if the mean adult male
pulse rate is 67.3 beats per minute with a standard
deviation of 10.3?
SOLUTION: Find the interval (48-2*10.3,48+2*10.3)
= (27.4, 68.6).
Is the data value in that interval? Yes. It is usual.
Finding the interval is a long way to test usual.
Today we learn a faster way called z scores.
Objective
We will use z scores to test whether values
are usual.
Measures of relative standing
Numbers showing the location of data values relative to
the other values within a data set.
These are used to
-compare values from different data sets
-compare values within the same data set.
The most important measure of relative standing is the z
score.
Also important are percentiles and quartiles, as well as a
new statistical graph called the boxplot.
Part 1
Basics of z Scores, Percentiles, Quartiles,
and Boxplots
z score
z Score (or standardized value)
The z score of a data value is the number of standard
deviations that the data value is away from the mean
Measures of Position z Score
Sample
xx
z
s
Population
z
x

Round z scores to 2 decimal places
Interpreting Z Scores
Whenever a value is less than the mean, its corresponding
z score is negative
Ordinary values: 2  z score  2
Unusual Values: z score  2 or z score  2
Example using z scores
The author of the text measured his pulse rate to
be 48 beats per minute.
Is that pulse rate unusual if the mean adult male
pulse rate is 67.3 beats per minute with a
standard deviation of 10.3?
x  x 48  67.3
z

 1.87
s
10.3
Answer: Since the z score is between – 2 and +2,
his pulse rate is not unusual.
Percentiles
are measures of location. Percentiles are
denoted P1, P2, . . ., P99, which divide a
set of data into 100 groups with about 1%
of the values in each group.
Given the Data Value
Find the Percentile
To find the
Percentile of a data
value x
the number of values less than x
total number of values
Count the number of values less than x
Divide by total number of values
Convert to a percent
EXAMPLE
For the 40 Chips Ahoy cookies, find the percentile for a cookie with
23 chips.
Answer: We see there are 10 cookies with fewer than 23 chips, so
10
Percentile of 23 
100  25
40
A cookie with 23 chips is in the 25th percentile.
Given the Percentile
Find the Data Value
Notation
k
L
n
100
1. Multiply the number of data
values*the percentile/100
2. Count up that many from the
bottom
percentile being used
locator that gives the position of
a value
Pk kth percentile
k
L
Converting from the
kth Percentile to the
Corresponding Data Value
Quartiles
Are measures of location, denoted Q1, Q2, and
Q3, which divide a set of data into four groups
with about 25% of the values in each group.
 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%.
 Q3
(Third quartile) separates the bottom
75% of sorted values from the top 25%.
Quartiles
Q1, Q2, Q3
divide sorted data values into four equal parts
25%
(minimum)
25%
25%
25%
Q1 Q2 Q3
(median)
(maximum)
Other Statistics
 Interquartile Range (or IQR):
 Semi-interquartile Range:
 Midquartile:
Q3  Q1
Q3  Q1
2
Q3  Q1
2
 10 - 90 Percentile Range: P90  P10
5-Number Summary
For a set of data, the 5-number summary consists of these five values:
1.
Minimum value
2.
First quartile Q1
3.
Second quartile Q2 (same as median)
4.
Third quartile, Q3
5.
Maximum value
Boxplot
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.
Boxplot - Construction
Find the 5-number summary.
Construct a scale with values that include the minimum and maximum
data values.
Construct a box (rectangle) extending from Q1 to Q3 and draw a line in
the box at the value of Q2 (median).
Draw lines extending outward from the box to the minimum and
maximum values.
Boxplots
Boxplots - Normal Distribution
Normal Distribution:
Heights from a Simple Random Sample of Women
Boxplots - Skewed Distribution
Skewed Distribution:
Salaries (in thousands of dollars) of NCAA Football Coaches
Part 2
Outliers and
Modified Boxplots
Outliers
An outlier is a value that lies very far away from the vast majority of the other
values in a data set.
Important Principles
 An outlier can have a dramatic effect on the
mean and 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.
Putting It All Together
 So far, we have discussed several basic tools
commonly used in statistics –

Context of data

Source of data

Sampling method

Measures of center and variation

Distribution and outliers

Changing patterns over time

Conclusions and practical implications
 This is an excellent checklist, but it should not
replace thinking about any other relevant factors.
Quiz
Solve the problem. Round results to the nearest hundredth.
1) Scores on a test have a mean of 66 and a standard
deviation of 9. Michelle has a score of 57. Convert
Michelle's score to a z-score.
2) Find the standard deviation and variance for the data 26
32 29 16 45 19.
Find the indicated measure.
3) The test scores of 40 students are listed below. Find P85.
30 35 43 44 47 48 54 55 56 57
59 62 63 65 66 68 69 69 71 72
72 73 74 76 77 77 78 79 80 81
81 82 83 85 89 92 93 94 97 98
4) Find the data value that corresponds to P30.
HW 123/1-31 odds 127,128 all problems