Chapter 2
Descriptive Statistics
Larson/Farber 4th ed.
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Chapter Outline
• 2.1 Frequency Distributions and Their Graphs
• 2.2 More Graphs and Displays
• 2.3 Measures of Central Tendency
• 2.4 Measures of Variation
• 2.5 Measures of Position
Larson/Farber 4th ed.
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Overview
Descriptive Statistics
• Describes the important characteristics of a set of
data.
• Organize, present, and summarize data:
1. Graphically
2. Numerically
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Important Characteristics of
Quantitative Data
“Shape, Center, and Spread”
• Center: A representative or average value that
indicates where the middle of the data set is located.
• Variation: A measure of the amount that the values
vary among themselves.
• Distribution: The nature or shape of the distribution
of data (such as bell-shaped, uniform, or skewed).
Overview
• 2.1 Frequency Distributions and Their Graphs
• 2.2 More Graphs and Displays
• 2.3 Measures of Central Tendency
• 2.4 Measures of Variation
• 2.5 Measures of Position
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Section 2.1
Frequency Distributions
and Their Graphs
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Frequency Distributions
Frequency Distribution
•
A table that organizes data values into classes or
intervals along with number of values that fall in
each class (frequency, f ).
1. Ungrouped Frequency Distribution – for data
sets with few different values. Each value is in
its own class.
2. Grouped Frequency Distribution: for data sets
with many different values, which are grouped
together in the classes.
Grouped and Ungrouped
Frequency Distributions
Ungrouped
Courses Frequency, f
Taken
Grouped
1
25
Age of Frequency, f
Voters
18-30
202
2
38
31-42
508
3
217
43-54
620
4
1462
55-66
413
5
932
67-78
158
6
15
78-90
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Ungrouped Frequency Distributions
Number of Peas in a Pea
Pod
Sample Size: 50
Peas per
pod
Freq, f
Freq,
Peas per pod
f
1
1
5
5
4
6
4
3
7
6
3
5
2
2
6
5
4
5
5
3
5
6
2
3
5
5
4
9
5
5
7
4
3
4
5
4
5
6
5
18
5
1
6
2
6
6
12
6
6
6
6
4
7
3
4
5
4
5
3
5
5
7
6
5
Graphs of Frequency Distributions:
Frequency Histograms
frequency
Frequency Histogram
• A bar graph that represents the frequency distribution.
• The horizontal scale is quantitative and measures the
data values.
• The vertical scale measures the frequencies of the
classes.
• Consecutive bars must touch.
data values
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Frequency Histogram
Ex. Peas per Pod
Number of Peas in a Pod
Freq, f
1
1
20
2
2
15
3
5
4
9
5
18
6
12
Frequency, f
Peas per pod
10
5
0
1
2
3
4
5
Number of Peas
7
3
6
7
Relative Frequency Distributions and
Relative Frequency Histograms
Relative Frequency Distribution
• Shows the portion or percentage of the data that falls
in a particular class.
• relative frequency

class frequency
Sample size

f
n
Relative Frequency Histogram
• Has the same shape and the same horizontal scale as
the corresponding frequency histogram.
• The vertical scale measures the relative frequencies,
not frequencies.
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Relative Frequency Histogram
Has the same shape and horizontal scale as a
histogram, but the vertical scale is marked with
relative frequencies.
Grouped Frequency Distributions
Grouped Frequency Distribution
• For data sets with many different values.
• Groups data into 5-20 classes of equal width.
Exam Scores
Freq, f
Exam Scores
Freq, f
30-39
30-39
1
40-49
40-49
0
50-59
50-59
4
60-69
60-69
9
70-79
70-79
13
80-89
80-89
10
90-99
90-99
3
Exam Scores
Freq, f
Grouped Frequency Distribution Terms
• Lower class limits: are the smallest numbers that
can actually belong to different classes
• Upper class limits: are the largest numbers that can
actually belong to different classes
• Class width: is the difference between two
consecutive lower class limits
15
Labeling Grouped Frequency
Distributions
• Class midpoints: the value halfway between LCL
and UCL:
(Low er class lim it)  (U pper class lim it)
2
• Class boundaries: the value halfway between an
UCL and the next LCL
(U pper class lim it)  (next Low er class lim it)
2
Constructing a Grouped Frequency
Distribution
1. Determine the range of the data:
 Range = highest data value – lowest data value
 May round up to the next convenient number
2. Decide on the number of classes.
 Usually between 5 and 20; otherwise, it may be
difficult to detect any patterns.
3. Find the class width:
range
 .class w idth =
num ber of classes
 Round up to the next convenient number.
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Constructing a Frequency Distribution
4. Find the class limits.
 Choose the first LCL: use the minimum data
entry or something smaller that is convenient.
 Find the remaining LCLs: add the class width to
the lower limit of the preceding class.
 Find the UCLs: Remember that classes must
cover all data values and cannot overlap.
5. Find the frequencies for each class. (You may add a
tally column first and make a tally mark for each
data value in the class).
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“Shape” of Distributions
Symmetric
•
Data is symmetric if the left half of its histogram is
roughly a mirror image of its right half.
Skewed
• Data is skewed if it is not symmetric and if it
extends more to one side than the other.
Uniform
• Data is uniform if it is equally distributed (on a
histogram, all the bars are the same height or
approximately the same height).
The Shape of Distributions
Symmetric
Skewed left
Uniform
Skewed Right
Outliers
Outliers
• Unusual data values as compared to the rest of the set.
They may be distinguished by gaps in a histogram.
Section 2.2
More Graphs and Displays
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Other Graphs
Besides Histograms, there are other methods of
graphing quantitative data:
•
•
•
Stem and Leaf Plots
Dot Plots
Time Series
Stem and Leaf Plots
Represents data by separating each data value into
two parts: the stem (such as the leftmost digit) and
the leaf (such as the rightmost digit)
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Constructing Stem and Leaf Plots
•
•
•
•
•
Split each data value at the same place value to form
the stem and a leaf. (Want 5-20 stems).
Arrange all possible stems vertically so there are no
missing stems.
Write each leaf to the right of its stem, in order.
Create a key to recreate the data.
Variations of stem plots:
1. Split stems
2. Back to back stem plots.
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Constructing a Stem-and-Leaf Plot
Include a key to identify
the values of the data.
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Dot Plots
Dot plot
• Consists of a graph in which each data value is
plotted as a point along a scale of values
Figure 2-5
Time Series
(Paired data)
Quantitative
data
Time Series
• Data set is composed of quantitative entries taken at
regular intervals over a period of time.
 e.g., The amount of precipitation measured each
day for one month.
• Use a time series chart to graph.
time
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Time-Series Graph
Number of Screens at Drive-In Movies Theaters
Figure 2-8
Ex. www.eia.doe.gov/oil_gas/petroleum/
Graphing Qualitative Data Sets
Pareto Chart
• A vertical bar graph in which the
height of each bar represents
frequency or relative frequency.
Frequency
Pie Chart
• A circle is divided into sectors
that represent categories.
Categories
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Constructing a Pie Chart
• Find the total sample size.
• Convert the frequencies to relative frequencies (percent).
Marital Status
Frequency,f Relative frequency (%)
(in millions)
Never Married
55.3
Married
127.7
Widowed
13.9
Divorced
22.8
Total: 219.7
55.3
 0.25 or 25%
219.7
127.7
219.7
13.9
219.7
22.8



219.7
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Constructing Pareto Charts
• Create a bar for each category, where the height of the
bar can represent frequency or relative frequency.
• The bars are often positioned in order of decreasing
height, with the tallest bar positioned at the left.
Figure 2-6
Section 2.3
Measures of Central Tendency
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Measures of Central Tendency
Measure of central tendency
• A value that represents a typical, or central, entry of a
data set.
• Most common measures of central tendency:
 Mean
 Median
 Mode
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Measure of Central Tendency: Mean
Mean : The sum of all the data entries divided by the
number of entries.
• Population mean:
 
x
N
• Sample mean:
x
x
n
• Round-off rule for measures of center:
Carry one more decimal place than is in the original
values. Do not round until the last step.
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Measure of Central Tendency: Median
Median
• The value that lies in the middle of the data when the
data set is arranged in order from lowest to highest. .
• Measures the center of an ordered data set by dividing
it into two equal parts.
• A sample mean is often referred to as ~
x.
• If the data set has an
 odd number of entries: median is the middle data
entry.
 even number of entries: median is the mean of
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ed. two middle data entries.
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the
Computing the Median
If the data set has an:
• odd number of entries: median is the middle data entry:
2
5
6
11
13
median is the exact middle value: x  6
• even number of entries: median is the mean of the two
middle data entries:
2
5
6
7
11
13
median is the mean of the by two numbers: x 
67
 6.5
2
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Measure of Central Tendency: Mode
Mode
• The data entry that occurs with the greatest frequency.
• If no entry is repeated the data set has no mode.
• If two entries occur with the same greatest frequency,
each entry is a mode (bimodal).
a) 5.40 1.10 0.42 0.73 0.48
 Mode is 1.10
1.10
 Bimodal -
b) 27 27 27 55 55 55 88 88 99
 No Mode
c) 1 2 3 6 7 8 9 10
27 & 55
Comparing the Mean, Median, and Mode
• All three measures describe an “average”. Choose the
one that best represents a “typical” value in the set.
• Mean:
 The most familiar average.
 A reliable measure because it takes into account
every entry of a data set.
 May be greatly affected by outliers or skew.
• Median:
 A common average.
 Not as effected by skew or outliers.
• Mode: May be used if there is an overwhelming repeat.
Choosing the “Best Average”
• The shape of your data and the existence of any
outliers may help you choose the best average:
Section 2.4
Measures of Variation
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Measures of Variation (“Spread”)
Another important characteristic of quantitative data
is how much the data varies, or is spread out.
The 2 most common method of measuring spread are:
1. Range
2. Standard deviation and Variance
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Range
Range
• The difference between the maximum and minimum
data entries in the set.
• The data must be quantitative.
• Range = (Max. data entry) – (Min. data entry)
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Example: Finding the Range
The wait time to see a bank teller is studied at 2 banks.
Bank A has multiple lines, one for each teller.
Bank B has a single wait line for 1st available teller.
5 wait times (in minutes) are sampled from each bank:
Bank A:
5.2 6.2 7.5 8.4 9.2
Bank B:
6.6 6.8 7.5 7.7 7.9
Find the mean, median, and range for each bank.
Solution: Finding the Range
• Bank A: Range = ?
• Bank B: Range = ?
• Note: The range is easy to compute, but only uses 2
values. Do the following 2 sets vary the same?
 Set A: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
 Set B: 1, 10, 10, 10, 10, 10, 10, 10, 10, 10
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Standard Deviation and Variance
Measures the typical amount data deviates from the
mean.
2
Sample Variance, s :
•
s 
2
(x  x )
2
n 1
Sample Standard Deviation, s:
•
s
s 
2
(x  x )
n 1
2
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Finding Sample Variance & Standard Deviation
x 
1. Find the mean of the sample
data set.
x
n
xx
2.
Find deviation of each entry.
3.
Square each deviation.
(x  x )
4.
Add to get the sum of the
deviations squared.
(x  x )
5.
Divide by n – 1 to get the
sample variance.
6.
Find the square root to get
the sample standard
deviation.
s 
2
s
2
2
(x  x )
2
n 1
(x  x )
2
n 1
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Find the Standard Deviation and Variance
for Bank A (multi-line)
x
x
36.5

n
s 
2
s
 7.3 m in
5
(x  x )
n 1
2

Wait time, Deviation: x – x
x (in min)
Squares: (x – x)2
5.2
5.2 – 7.3 = -2.1
(–2.1)2 = 4.41
6.2
6.2 – 7.3 =
(
)2 =
7.5
7.5 – 7.3 =
(
)2 =
8.4
8.4 – 7.3 =
(
)2 =
9.2
9.2 – 7.3 =
(
)2 =
 x  36.5
Σ(x – x) =
x  x 
2
s 
2
• Round to one more decimal than the data.
• Don’t round until the end.
• Include the appropriate units.
Find the Standard Deviation and Variance
for Bank B (1 wait line)
x
x
36.5

n
 7.3 m in
5
Wait time, Deviation: x – x
x (in min)
Squares: (x – x)2
6.6
6.8
s 
2
(x  x )
n 1
7.5
2

7.7
7.9
 x  36.5
s
Σ(x – x) =
x  x 
2
s 
2
• Round to one more decimal than the data.
• Don’t round until the end.
• Include the appropriate units.
Sample versus Population
Standard Deviation and Variance
Sample
Statistics:
Population
Parameters:
Mean
x
µ
Standard
Deviation
s
σ
Variance
s2
σ2
Sample versus Population
Standard Deviation
Note: Unlike x and µ, the formulas for s and σ
are not mathematically the same:
Sample Standard Deviation
•
s
s 
2
(x  x )
2
n 1
Population Standard Deviation
•
 
Larson/Farber 4th ed.

2

(x   )
2
N
51
Standard Deviation: Key Points

s 0
( When would s = 0 ?)
 The standard deviation is a measure of variation of all
values from the mean. The larger s is, the more the
data varies.
 The units of the standard deviation s are the same as
the units of the original data values. (The variance
has units2).
 The value of the standard deviation s can increase
dramatically with the inclusion of one or more
outliers (data values far away from all others)
Interpreting Standard Deviation
• Standard deviation is a measure of the typical amount
an entry deviates from the mean.
• The more the entries are spread out, the greater the
standard deviation.
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Solution: Using Technology to Find the
Standard Deviation
Sample Mean
Sample Standard
Deviation
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Using Technology
The gas mileage of 2 cars is sampled over various
conditions:
Car A:
Car B:
21.1 21.2 20.8 19.8 23.8 (mpg)
25.2 19.1 18.0 24.4 20.3 (mpg)
Which car do you think gets “better” mpg?
Use a calculator to find the mean and standard deviation
for each to justify your choice.
Standard Deviation and “Spread”
How does “s” show how much the data varies?
Three methods:
1. Range Rule of Thumb
2. Chebyshev’s Theorem
3. The Empirical Rule
The Range Rule of Thumb
Range Rule: For most data sets, the majority of the
data lies within 2 standard deviations of the mean.
Recall: Range = High – Lo
Estimate: Range ≈ 4s
Alternatively, If the range is known, you can use the range
rule to estimate the standard deviation:
s
Range
4
Using the Range Rule of Thumb
A sample of women’s heights has a mean of 64
inches and a standard deviation of 2.5 inches.
Using the range rule, “most” women fall within
what heights?
What would be an “unusual” height?
Using the Range Rule of Thumb
The sample of Exam Scores used in the class
handout had a mean of 73.6. Which of the
following is most likely the standard deviation of
the sample?
s = 3.6
s = 12.8
s = 74.5
Use the range rule to help justify your choice.
Chebyshev’s Theorem
Chebyshev’s Theorem
For data with any distribution, the proportion (or
fraction) of any set of data lying within K standard
deviations of the mean is always at least 1-1/K2, where
K is any positive number greater than 1.
 For K = 2, at least 3/4 (or 75%) of all values lie
within 2 standard deviations of the mean
 For K = 3, at least 8/9 (or 89%) of all values lie
within 3 standard deviations of the mean
Using Chebyshev’s Theorem
A sample of salaries at an elementary school has a
mean of $32,000 and a standard deviation of $3000.
Use Chebyshev’s Theorem to describe how the salaries
are spread out.
Would a salary of $28,000 be “unusual?”
Would a salary of $45,000 be “unusual”?
The Empirical Rule
Empirical (68-95-99.7) Rule
For data sets having a symmetric distribution:
 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
The Empirical Rule
Example: Using the Empirical Rule
A sample of IQs has a symmetric distribution with a mean
of 100 and a standard deviation of 15.
1. Sketch the distribution.
2. 68% of people have an IQ between what 2 values?
3. What percent of people have an IQ between 70 and 130?
4. What percent of people have an IQ between 100 and 115?
5. What percent of people have an IQ above 145?
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