Numerical Descriptive Measures

1

• Chapter 3
• Numerical Descriptive Measures

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Summary Measures
Central Tendency
Variation
Mean
Mode
Quartile
Median
Range
Variance
Standard Deviation
Geometric Mean
Coefficient of Variation
3
Measures of Central Tendency

• Various ways to describe the central, most common
  or middle value in a distribution or set of data
• The Mean (Arithmetic Mean)
• The Median
• The Mode
• The Geometric Mean

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The Mean

• Equals the sum of all observations or values
  divided by the number of values
• The Most Common Measure of Central Tendency
• Generally called the Average in common usage
• Population mean vs. Sample mean

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The Mean

• Population mean µ (mu)
• Recall N population size
• Sample mean X (x-bar)
• Recall n sample size
• S Sum
• Sx Sum of all values of x

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The Mean

• Affected by Extreme Values (Outliers)
• Note how a difference of one value affects the
  mean in the example below

0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10 12
14
Mean 5
Mean 6
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The Median

• The Middle number
• The most Robust measure of Central Tendency
• Not affected by extreme values
• In an ordered array, the Median (n1)/2 ordered
  observation.
• If n or N is odd, the median is the middle number
• If n or N is even, the median is the average of
  the 2 middle numbers

0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10 12
14
Median 5
Median 5
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Mode

• The Value that Occurs Most Often
• Not Affected by Extreme Values
• There may be gt1 Modes
• There may be no Mode
• Can be used for Numerical or Categorical Data

0 1 2 3 4 5 6
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14
No Mode
Mode 9
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Example
Jim has 20 problems to do for homework. Some are
harder than others and take more time to solve.
We take a random sample of 9 problems. Find the
mean, median and mode for the number of minutes
Jim spends on his homework.
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Solution Mean
Sample size (n) 9 Problems 1 through 9 x1,
x2, x3 x9, respectively. Sx (12 4 3 8
7 5 4 9 11) 63 minutes Sx/n 63/9
7 minutes
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Solution Median
Place the data in ascending order as at
right. (n1)/2 (91)/2 5 The 5th ordered
observation is 7 and so is the Median.
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Solution Mode
Since the data is already arranged in order from
smallest to largest we will keep it that
way. Only the value 4 occurs gt1 time. The Mode
is 4.
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Solution Excel/PHStat

• Equations
• Mean (AVERAGE)
• Median (MEDIAN)
• Mode (MODE)
• Excel worksheet

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Approximating the Mean from a Frequency
Distribution

• Used when the only source of data is a frequency
  distribution

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Example

• X ((153) (256) (355) (454)
  (552))/20
• (45 150 175 180 110)/20
• 660/20
• 33

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Measures of Variation
Variation
Variance
Standard Deviation
Coefficient of Variation
Range
Population Variance
Population Standard Deviation
Sample Variance
Sample Standard Deviation
Interquartile Range
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Range

• Difference between the Largest and the Smallest
  Observations
• Ignores the distribution of data

Range 12 - 7 5
Range 12 - 7 5
7 8 9 10 11 12
7 8 9 10 11 12
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Quartiles

• Quartiles Split Ordered Data into 4 equal
  portions
• Q1 and Q3 are Measures of Non-central Location
• Q2 the Median

25
25
25
25
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Quartiles

• Each Quartile has position and value
• With the data in an ordered array, the position
  of Qi is
• The value of Qi is the value associated with that
  position in the ordered array
• Example

Data in Ordered Array 11 12 13 16 16
17 18 21 22
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Quartiles Example
Find the 1st and 3rd Quartiles in the ordered
observations at right. Position of Q1 1(91)/4
2.5 The 2.5th observation (44)/2 4
Position of Q3 3(91)/4 3(Q1) 7.5 The
7.5th observation (911)/2 10
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Interquartile Range (IQR)

• The difference between Q1 and Q3
• The middle 50 of the values
• Also Known as Midspread
• Resistant to extreme values
• Example
• Q1 12.5, Q3 17.5
• 17.5 12.5 5
• IQR 5

11 12 13 16 16 17 17 18 21
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Range and IQR Example
Find the Range and the Interquartile Range in
this distribution. Range largest smallest
12 3 9. Position of Q1 1(91)/4 2.5 The
2.5th observation (44)/2 4 Position of Q3
3(91)/4 3(Q1) 7.5 The 7.5th observation
(911)/2 10 IQR 10 4 6
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Variance

• Shows Variation about the Mean
• Population Variance (s2)
• Sample Variance (S2)

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Standard Deviation (SD)

• Most Important Measure of Variation
• The square root of the variance
• Shows Variation about the Mean
• Population Standard
  Deviation (s)
• Sample Standard Deviation (S)

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Variance and Standard Deviation

• Both measure the average scatter about the mean
• Variance computations produce squared units
  which makes interpretation more difficult
• For example, 2 is meaningless.
• Since it is the square root of the Variance, the
  Standard Deviation is expressed in the same units
  as the original data
• Therefore, the Standard Deviation is the most
  commonly used measure of variation

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A trivial Standard Deviation Example
Sample values 3, 4, 5 Sample mean 4 n 3

• S (3-4)2 (4-4)2 (5-4)2
• 3-1
• S -12 02 12
• 2
• S 2/2 1

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Comparing Standard Deviations

• Greater S (or s) more dispersion of data

Data A
Mean 15.5 s 3.338
11 12 13 14 15 16 17 18
19 20 21
Data B
Mean 15.5 s .9258
11 12 13 14 15 16 17 18
19 20 21
Data C
Mean 15.5 s 4.57
11 12 13 14 15 16 17 18
19 20 21
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Approximating SD from a Frequency Distribution

• Used when the raw data are not available and the
  only source of data is a frequency distribution

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Approximating SD from a Frequency Distribution
X 33 S2 (3(15-33)2 6(25-33)2 5(35-33)2
4(45-33)2 2(55-33)2)/ 20-1 S2 (972 384 20
576 968)/19 2920/19 S2 153.7 S 12.4
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PhStat

• Dispersion
• Qi QUARTILE
• S2 VAR
• S STDEV
• s2 VARP
• s STDEVP
• Excel worksheet

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Coefficient of Variation

• Measure of Relative Variation
• Shows Variation Relative to the Mean
• Used to Compare Two or More Sets of Data Measured
  in Different Units
• S Sample Standard Deviation
• X Sample Mean

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Comparing Coefficientof Variation

• Stock A
• Average price last year 50
• Standard deviation 2
• Stock B
• Average price last year 100
• Standard deviation 5
• Coefficient of Variation
• Stock A
• Stock B

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The Z - Score

• Z Score difference between the value and the
  mean, divided by the standard deviation
• Useful in identifying outliers (extreme values)
• Outliers are values in a data set that are
  located far from the mean
• The larger the Z Score, the larger the distance
  from the mean
• A Z Score is considered an outlier if it is lt
  -3.0 or gt 3.0
• Formulae

Z X - X S
Z X - µ s
OR
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Z Score Example

• If our sample mean 25, and our sample standard
  deviation is 10, is a value of 65 an outlier?
• 4.0 gt 3.0, so 65 is an outlier.

Z X - X S
65 - 25 40 4.0 10 10
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Shape of a Distribution
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The Empirical Rule

• For most data sets
• 68 of the Observations Fall Within () 1
  Standard Deviation Around the Mean
• 95 of the Observations Fall Within () 2 SD
  Around the Mean
• 99.7 of the Observations Fall Within () 3 SD
  Around the Mean
• More accurate for fairly symmetric data sets

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The Empirical Rule
68 of the Observations Fall Within () 1
Standard Deviation Around the Mean
68
-1sd µ 1sd
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The Empirical Rule
95 of the Observations Fall Within () 2 SD
Around the Mean
95
-2sd -1sd µ 1sd
2sd
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The Empirical Rule
99.7 of the Observations Fall Within () 3 SD
Around the Mean
99.7
-3sd -2sd -1sd µ 1sd
2sd 3sd
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The Bienayme-Chebyshev Rule

• The Percentage of Observations Contained Within
  Distances of k Standard Deviations Around the
  Mean Must Be at Least
• Applies regardless of the shape of the data set

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The Bienayme-Chebyshev Rule

• At least () 75 of the observations must be
  contained within distances of 2 SD around the
  mean
• At least () 88.89 of the observations must be
  contained within distances of 3 SD around the
  mean
• At least () 93.75 of the observations must be
  contained within distances of 4 SD around the mean

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The Bienayme-Chebyshev Rule
75
- 2sd µ 2sd
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The Bienayme-Chebyshev Rule
75
88.89
- 3sd - 2sd µ 2sd
3sd
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The Bienayme-Chebyshev Rule
75
88.89
93.75
- 4sd - 3sd - 2sd µ
2sd 3sd 4sd
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Shape of a Distribution

• Describe How Data are Distributed
• Measures of Shape
• Symmetric or Skewed (asymmetric)

Right-Skewed
Left-Skewed
Symmetric
Mean lt Median lt Mode
Mean Median Mode

Mode lt Median lt Mean
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The Box Plot (Box-and-Whisker)

• 5 number summary
• Median, Q1, Q3, Xsmallest, Xlargest
• Box Plot
• Graphical display of data using 5-number summary

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4
6
8
10
X
largest
X
Median
smallest
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Distribution Shape Box Plot
Right-Skewed
Left-Skewed
Symmetric
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Correlation Coefficient

• Correlation Coefficient r
• Unit Free
• Measures the strength of the linear relationship
  between 2 quantitative variables
• Ranges between 1 and 1
• The Closer to 1, the stronger the negative
  linear relationship becomes
• The Closer to 1, the stronger the positive linear
  relationship becomes
• The Closer to 0, the weaker any linear
  relationship becomes

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Scatter Plots of Data with Various Correlation
Coefficients
Y
Y
Y
X
X
X
r -1
r -.6
r 0
Y
Y
X
X
r 1
r .6
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Pitfalls in Numerical Descriptive Measures and
Ethical Issues

• Data Analysis is Objective
• Should report the summary measures that best meet
  the assumptions about the data set
• Data Interpretation is Subjective
• Should be done in a fair, neutral and clear
  manner
• Ethical Issues
• Should document both good and bad results
• Presentation should be fair, objective and
  neutral
• Should not use inappropriate summary measures to
  distort the facts