Descriptive Statistics: An
Overview of Statistics and
Probability | STAT 211
Statistics
Texas A&M University (A&M)
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Statistics 211
✬
✩
[Student]
1 An Overview of Probability and Statistics
1.1 What is Statistics?
In short: Analysis of data (all stages).
Common perceptions of statistics:
The above are “descriptions of the world” using numbers. This is a part
of Statistics (very visible), but statistics deals with more than “describing
phenomena”
Examples of Statistics:
Polio Vaccine In the 1950’s Polio was a serious disease that affected
countless people (mostly children). In 1954
• 401,974 children vaccinated.
• 201,229 with a trial vaccine and
• 200,745 with a placebo.
There where a total of
: Polio cases.
for placebo versus
for vaccine.
Question:
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Chapter 1: Descriptive Statistics
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Slide 1
Statistics 211
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[Student]
Unemployment We desire to know the unemployment rate.
Problem:
How do you find the answer?
How accurate are the results?
Stress Traffic lights are installed to aid in merging into the Interstate
(I-75 in Tampa, FL).
• Stress level of drivers is measured before the lights:
• After the lights it was
.
.
Question:
1.2 Branches of Statistics
Descriptive (deductive) statistics. Statistical methods that summarize
and describe the prominent features of data.
Inferential (inductive) statistics. Statistical methods that generalize from
a sample to a population.
Population The entire collection of individuals objects or
measurements about which information is desired.
Sample A part or subset of the population.
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Statistics 211
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[Student]
Examples:
• Polio:
• Unemployment:
• Stress:
A population or sample is not static, but depends upon the definition of
the problem.
Most of the samples in this course will be
—a sample
that is randomly chosen from the population.
Historically
statistics were far more important than
statistics. Nowadays the reverse is true.
1.3 Data Definitions
Categorical vs. Numerical
Categorical Observations that are only classified into groups.
Examples:
Numerical Observations that have a numerical quality about them.
Examples:
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Slide 3
Statistics 211
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Classify the following:
State of birth
Weight on birth
Date of birth
Zip code
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[Student]
Discrete vs. Continuous
Discrete A variable is discrete if it can assume only a countable
number of possible values.
Examples:
Continuous A variable is continuous if it can assume an
uncountable number of values.
Examples:
There will usually be practical limitations on the accuracy any
continuous variables has.
Data Sets
Univariate Data set consists of
Bivariate . . .
variable.
variables.
Multivariate . . . more than
variables.
Example of a multivariate data set:
✫
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Statistics 211
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[Student]
2 Pictorial and Tabular Methods in
Descriptive Statistics
Consider the Following Data Set:
(Chp 1, #10) The concentration of suspended solids in river water is an
important environmental characteristic. The paper “Water Quality in
Agricultural Watershed: Impact of Riparian Vegetation During Base Flow”
(Water Resources Bull., 1981, pp. 233-239) reported on concentrations
(in parts per million, or ppm) for several different rivers. Suppose the
following 50 observations had been obtained for a particular river.
55.8
45.9
83.2
75.3
60.7
60.9
39.1
40.0
71.4
77.1
37.0
35.5
31.7
65.2
59.1
91.3
56.0
36.7
52.6
49.5
65.8
44.6
62.3
58.2
69.3
42.3
71.7
47.3
48.0
69.8
33.8
61.2
94.6
61.8
64.9
60.6
61.5
56.3
78.8
27.1
76.0
47.2
30.0
39.8
87.1
69.0
74.5
68.2
65.0
66.3
Question: What does this data tell us about the concentration of
suspended solids?
First few steps in analyzing a data set:
1. Organize and summarize the data.
2. Find the center of the data.
3. Examine the spread of the data.
✫
Chapter 1: Descriptive Statistics
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Statistics 211
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[Student]
2.1 Stem and Leaf Display
A compact and descriptive method of organizing data without losing any
information in the data.
• Leading digits are stems
• Trailing digits are leaves.
• Indicate units somewhere on the display.
• Option: Sort the leaves.
• Comparative stem & leaf.
• Repeat stems if need be.
Advantages:
• No loss of information.
• Easy to do for small data sets.
Disadvantages:
• Time consuming for large data sets (by hand)
• Cannot be used for categorical data.
• Very space consuming for large data sets.
✫
Chapter 1: Descriptive Statistics
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Slide 6
Statistics 211
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[Student]
A Stem-and-leaf display of the solids data set:
55.8
45.9
83.2
75.3
60.7
60.9
39.1
40.0
71.4
77.1
37.0
35.5
31.7
65.2
59.1
91.3
56.0
36.7
52.6
49.5
65.8
44.6
62.3
58.2
69.3
42.3
71.7
47.3
48.0
69.8
33.8
61.2
94.6
61.8
64.9
✫
Chapter 1: Descriptive Statistics
60.6
61.5
56.3
78.8
27.1
76.0
47.2
30.0
39.8
87.1
69.0
74.5
68.2
65.0
66.3
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Slide 7
Statistics 211
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Stem-and-leaf display of the solids data set with sorted leaves:
2
3
4
5
6
7
8
9
:
:
:
:
:
:
:
:
7
0245779
002567789
366689
111112255566899
01245679
37
15
✩
[Student]
units: ppm
Stem-and-leaf display with multiple leaf values on a stem:
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
7
024
5779
002
567789
3
66689
1111122
55566899
0124
5679
3
7
1
5
units: ppm
✫
Chapter 1: Descriptive Statistics
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Slide 8
Statistics 211
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[Student]
Comparative Stem-and-leaf display on the solids data set taken two
years earlier:
Two Years
Current
------------------------------------8 : 1 :
9851 : 2 : 7
9887640 : 3 : 0245779
9997765322111 : 4 : 002567789
877554200 : 5 : 366689
9887653221 : 6 : 111112255566899
72210 : 7 : 01245679
95 : 8 : 37
: 9 : 15
units: ppm
Sometimes we redefine the leaves for low-numbered or narrow data sets:
58, 58, 57, 54, 54, 54, 57, 57, 56, 56, 57, 51, 58, 54, 52, . . . , 52, 54
60
59
58
57
56
55
54
53
52
51
:
:
:
:
:
:
:
:
:
:
0
00
00000000000
0000000000
0000000000
0000000000000
0000000000000
0000
000
0
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Slide 9
Statistics 211
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✩
[Student]
2.2 Frequency Distributions for Quantitative Data
A very popular way to summarize data is with a frequency distribution.
A frequency distribution is a compact summary of a data set using a
table with 3 or 4 columns:
Class interval (or category) disjoint intervals of each obs in the
data set.
Frequency Number of obs in a class interval =
f.
Relative frequency Proportion of obs in interval =
f /n
Cumulative frequency Sum of the relative frequencies
Pclass
i=1
f /n.
Number of classes: 5 to 20. Use
√
n for a rough idea.
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Slide 10
Statistics 211
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[Student]
A Frequency Distribution for the solids data set:
55.8
45.9
83.2
75.3
60.7
60.9
39.1
40.0
71.4
77.1
37.0
35.5
31.7
65.2
59.1
91.3
56.0
36.7
52.6
49.5
65.8
44.6
62.3
58.2
69.3
42.3
71.7
47.3
48.0
69.8
33.8
61.2
94.6
61.8
64.9
60.6
61.5
56.3
78.8
27.1
76.0
47.2
30.0
39.8
87.1
69.0
74.5
68.2
65.0
66.3
50 observations. Approximate number of classes:
Class Interval
[Tally]
Frequency
Relative f
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Chapter 1: Descriptive Statistics
Cumulative f
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Slide 11
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✩
[Student]
2.3 Histogram
A histogram is a pictorial representation of a frequency distribution.
1. Draw a x-axis and mark class intervals.
2. Draw a rectangle whose area is proportional to the frequency
0
5
10
15
of that interval.
20
40
60
80
100
solids
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Slide 12
Statistics 211
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✩
[Student]
A true histogram or a density scale will have an area that is equal to 1.0.
In that case we make the:
Rectangle Height =
Relative Frequency
Base Length
In the case where all the intervals are of equal length all we need to do
0.0
0.005
0.010
0.015
0.020
0.025
0.030
is add the appropriately labeled y-axis.
20
40
60
80
100
solids
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Slide 13
Statistics 211
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Histograms often exhibit particular shapes:
✩
[Student]
• unimodal
• bimodal
• multimodal
• symmetric
• positively skewed
• negatively skewed
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Slide 14
Statistics 211
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2.4 The M&M Data Set
✩
[Student]
Some important questions:
How many M&M’s are their in a regular plain size M&M bag?
More importantly, how many red M&M’s are there?
HW assignment: Buy a M&M bag (small, plain) and count the number
of M&M’s and the number of red M&M’s. Email them to me at
“henrik@stat.tamu.edu”.
Part of homework #1 assignment (red M&M’s for 1-3):
1. Create a Stem-and-Leaf plot.
2. Create a frequency distribution.
3. Plot a histogram (density scale).
4. Create a Comparative Stem-and-Leaf plot of the total number
of M&M’s (I will post the data on thew web) to that of Spring
1998 (next slide). Do you think the total number of M&M’s per
bag has changed?
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Slide 15
Statistics 211
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✩
[Student]
2.4.1 The M&M Data set for Spring 1998
The following data was collected by the Spring 1998 Stat 211 class:
Total M&M’s (n
58
57
58
56
58
60
55
Red M&M’s (n
18
14
18
12
15
11
13
= 68):
58
51
55
55
55
55
56
57
58
56
55
56
58
58
54
54
52
57
55
54
59
54
52
57
58
55
55
54
54
54
55
54
57
54
53
57
55
57
53
56
56
52
57
54
58
53
54
57
54
56
56
59
55
54
53
56
56
58
57
58
55
6
9
4
14
10
10
5
6
5
15
17
10
7
3
7
8
9
7
14
12
9
15
10
14
6
12
13
14
8
16
6
9
5
12
11
20
9
8
11
12
13
15
13
19
11
= 66):
16
2
11
10
10
15
19
13
15
12
11
11
7
8
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Slide 16
Statistics 211
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[Student]
3 Measures of Location
Another step in gaining understanding of our data is to find the “center”
of our data. What is the center?
3.1 Mean / Average
Average: If we consider each number to have a “weight” equal
to its value, then the average is the value which equally
divides the data by weight. Think of a seesaw:
We calculate the average as follows:
Sample Average
x̄ =
1
n
Pn
i=1 xi
Population Average
µ=
1
N
PN
xi :
The i’th observation in the sample.
yj :
The j ’th value in the population.
n:
Sample size.
N:
Note:
✫
Chapter 1: Descriptive Statistics
j=1
yj
Population sample size.
x1 , . . . , xn is a sample from population y1 , . . . , yN .
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Slide 17
Statistics 211
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✩
[Student]
Example: Calculate the average number of red M&M’s for the Spring
1998 M&M data set.
Red M&M’s (n
18
14
18
12
15
11
13
= 66):
16
2
11
10
10
15
19
13
15
12
11
11
7
8
6
9
4
14
10
10
5
x x x
x x x x
x x x x
x x x x x x x
x
x
x
x
x
6
5
15
17
10
7
3
x
x
x
x
x
x
x
x
x
x
x
x
x
7
8
9
7
14
12
9
x
x
x
x
x
x
x
x
x
x
x
15
10
14
6
12
13
x
x
x
x
x
14
8
16
6
9
5
x
x
x
x
x
x
x
x
12
11
20
9
8
11
x
x
x
12
13
15
13
19
11
x
x
x
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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Slide 18
Statistics 211
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✩
[Student]
3.2 Median
Median: The middle observation of the sorted data set.
Sample Median =
x̃
Population Median =
µ̃
We calculate the median:
n odd:
n even:
x̃ = x[(n+1)/2]
x̃ = (xn/2 + x[(n+2)/2] )/2
Example: Calculate the median number of red M&M’s for the Spring
1998 M&M data set.
Red M&M’s (n
2
7
9
11
12
14
17
= 66):
3
7
9
11
12
14
18
4
7
9
11
13
15
18
5
7
10
11
13
15
19
5
8
10
11
13
15
19
5
8
10
11
13
15
20
6
8
10
12
13
15
6
8
10
12
14
15
6
9
10
12
14
16
6
9
11
12
14
16
Discussion:
• Mean and for different types of “smoothed histograms”
(distributions) [slide 14].
• How do outliers affect the mean and median?
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Slide 19
Statistics 211
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[Student]
3.3 Other Measures of Location
3.3.1 Trimmed Mean
A trimmed mean is a compromise between x̄ and x̃ in that outliers will
have some effect on the trimmed mean but not as much as they have on
the mean. It is calculated by eliminating a certain percentage of the
observation from both ends and calculating the average of the remaining
data. For example a 10% trimmed mean would eliminate 10% of the
observation from each end of the data (20% total) and average the
remaining 80% of the observations.
For example: If we have a sample of 100 observation and we want to
find x̄12% (12% trimmed mean), how many observations
must we eliminate from each end?
Solution: We have n
= 100 observations. 12% of this is
100 × .12 = 12. Therefore we eliminate 12 observation from
each end for a total of 24 observations.
There are a variety of ways to handle the case where we need to chop of
a fractional number of data points. In this case we avoid the issue and
simply round the number of observations that are removed.
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Slide 20
Statistics 211
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✩
[Student]
Example: Calculate the 10% trimmed mean of red M&M’s for the Spring
1998 M&M data set.
Red M&M’s (n
2
7
9
11
12
14
17
= 66):
3
7
9
11
12
14
18
4
7
9
11
13
15
18
5
7
10
11
13
15
19
5
8
10
11
13
15
19
5
8
10
11
13
15
20
6
8
10
12
13
15
6
8
10
12
14
15
✫
Chapter 1: Descriptive Statistics
6
9
10
12
14
16
6
9
11
12
14
16
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Slide 21
Statistics 211
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✩
[Student]
3.3.2 Percentiles and Quartiles
The p’th percentile is the observation in our data set where p% are equal
to or less than this observation. The median is the 50’th percentile. To
calculate the p’th percentile x[p] :
1. Let x(i) refer to our data set in ascending order.
2. Let ip
= np/100.
3. Find the first index i such that i
> ip .
4. The p’th percentile is then:
x[p] =


x(i−1) +x(i)
2
 x(i)
if i − 1
= ip
otherwise
Q1 :
First Quartile
= 25’th percentile
= lower fourth.
Q2 :
Second Quartile
= 50’th percentile
= median.
Q3 :
Third Quartile
= 75’th percentile
= upper fourth
IQR = fs = Q3 − Q1 = “Interquartile Range” or “Fourth Spread.”
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Slide 22
Statistics 211
✬
Example: Calculate Q1 and Q3 for our Spring 1998 M&M data set.
Red M&M’s (n
2
7
9
11
12
14
17
= 66):
3
7
9
11
12
14
18
4
7
9
11
13
15
18
5
7
10
11
13
15
19
5
8
10
11
13
15
19
5
8
10
11
13
15
20
6
8
10
12
13
15
6
8
10
12
14
15
✫
Chapter 1: Descriptive Statistics
✩
[Student]
6
9
10
12
14
16
6
9
11
12
14
16
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Slide 23
Statistics 211
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✩
[Student]
3.3.3 Boxplots
Box plots are useful in summarizing various aspects of the data.
Side-by-side box plots provide useful comparisons of two or more sets of
data.
1. Form an axis that includes all possible values of the data.
2. Draw a box extending from Q1 to Q3 .
3. Draw a vertical bar at the median.
4. Draw whiskers (horizontal lines) out 1.5 IQR from each end of
the box.
5. Indicate mild outliers with a “◦” (1.5 − 3.0 IQR from each
end of the box).
6. Indicate extreme outliers with a “∗” (more than 3.0 IQ from
each end of the box).
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Slide 24
Statistics 211
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Example: Calculate the summary statistics x̄, x̃, Q1, Q3 for the water
✩
[Student]
quality data set. Then construct a box plot.
x̄ =
Min =
Q1 =
X̃ =
Q3 =
Max =
60
40
solids(ppm)
80
Particulate Matter
✫
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Statistics 211
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[Student]
3.3.4 Categorical Data and Sample Proportions
We cannot calculate mean and median for categorical data. However we
can calculate a sample proportion. We calculate the sample proportion:
Proportion = p =
Count
Sample Size
For example: What proportion—on the average—of M&M’s are red in for
the Spring 1998 M&M data set?
x̄red = 11.06
x̄total = 55.61
✫
Chapter 1: Descriptive Statistics
✪
c 1998-2004 by Henrik Schmiediche
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Statistics 211
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[Student]
4 Measures of Variability
The mean, median, etc. do not give us a complete overview (summary)
of our data.
For Example: Consider the following three data sets:
Data
1:
20
30
40
50
60
70
50
350
18.71
2:
20
43
44
46
47
70
50
252
15.87
3:
40
43
44
46
47
50
10
12
3.46
– The mean and median is 45 for all three data sets.
– These data sets have very different spreads.
Ways to measure spread:
Range: range = maximum observation – minimum observation
Average the Deviations from the Mean: We define the i′ th
deviation to be:
xi − x̄. Intuitive: We average the deviations:
1X
(xi − x̄)
n
Problem: this does not give us anything useful!
✫
Chapter 1: Descriptive Statistics
✪
c 1998-2004 by Henrik Schmiediche
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Statistics 211
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[Student]
Variance: When we average the squared deviations from the
mean and divide by n − 1 instead of n we get a measure of
spread we call the variance:
1 X
s =
(xi − x̄)2
n−1
2
Calculation formula:
s2 =
1
n−1
³X
x2i −
¡P
xi
n
– The population variance is represented as:
¢2 ´
σ2 .
– We will learn later why we divide by n–1 instead of n.
Standard Deviation: The units of the variance are units of the
data squared. To make the units the same as that of the data
set we take the square root of the variance. This is called the
standard deviation:
s=
√
s2
s is translation invariant:
s(x1 , ..., xn ) = s(x1 + a, ..., xn + a) ∀ a.
s is scale equivariant:
s(ax1 , ..., axn ) = |a|s(x1 , ..., xn ) ∀ a.
The population standard deviation is:
σ.
✫
Chapter 1: Descriptive Statistics
✪
c 1998-2004 by Henrik Schmiediche
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Statistics 211
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[Student]
Example: Calculate the range, variance and standard deviation of red
M&M’s for the Spring 1998 M&M data set. Remember:
x̄ = 11.06
Red M&M’s (n
18
14
18
12
15
11
13
= 66):
16
2
11
10
10
15
19
13
15
12
11
11
7
8
6
9
4
14
10
10
5
6
5
15
17
10
7
3
7
8
9
7
14
12
9
15
10
14
6
12
13
14
8
16
6
9
5
✫
Chapter 1: Descriptive Statistics
12
11
20
9
8
11
12
13
15
13
19
11
✪
c 1998-2004 by Henrik Schmiediche
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