Download Basic Business Statistics, 8th Edition

Numerical Descriptive
Measures
Chapter 2
Borrowed from
http://www2.uta.edu/infosys/amer/courses/3
321%20ppts/c3.ppt
Chapter Topics

Measures of central tendency

Mean, median, mode, midrange

Quartiles

Measure of variation


Shape


Range, interquartile range, variance and Standard
deviation, coefficient of variation
Symmetric, skewed (+/-)
Coefficient of correlation
Summary Measures
Summary Measures
Central Tendency
Arithmetic
Mean
Quartile
Median Mode
Range
Variation
Coefficient of
Variation
Variance
Geometric Mean
Standard Deviation
Measures of Central Tendency
Central Tendency
Average (Mean)
Median
n
X 
X
i 1
n
N

X
i 1
N
i
i
Mode
Mean (Arithmetic Mean)

Mean (arithmetic mean) of data values

Sample mean
Sample Size
n
X

X
i 1
i
n
X1  X 2 

n
Population mean
N

X
i 1
N
i
 Xn
Population Size
X1  X 2 

N
 XN
Mean (Arithmetic Mean)
(continued)


The most common measure of central
tendency
Affected by extreme values (outliers)
0 1 2 3 4 5 6 7 8 9 10
Mean = 5
0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 6
Median


Robust measure of central tendency
Not affected by extreme values
0 1 2 3 4 5 6 7 8 9 10
Median = 5

0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5
In an Ordered array, median is the “middle”
number


If n or N is odd, median is the middle number
If n or N is even, median is the average of the two
middle numbers
Mode






A measure of central tendency
Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may may be no mode
There may be several modes
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Mode = 9
0 1 2 3 4 5 6
No Mode
Quartiles

Split Ordered Data into 4 Quarters
25%
25%
 Q1 

25%
 Q2 
Position of i-th Quartile
25%
Q3 
i  n  1
 Qi  
4
Data in Ordered Array: 11 12 13 16 16 17 18 21 22
1 9  1
Position of Q1 
 2.5
4
Q1
12  13


 12.5
2
Q1 and Q3 Are Measures of Noncentral Location
 Q = Median, A Measure of Central Tendency
2

Measures of Variation
Variation
Variance
Standard Deviation
Range
Interquartile
Range
Population
Variance (σ2)
Sample
Variance (S2)
Coefficient
of Variation
Population
Standard deviation
(σ)
Sample Standard
deviation (S)
Range


Measure of variation
Difference between the largest and the
smallest observations:
Range  X Largest  X Smallest

Ignores the way in which data are distributed
Range = 12 - 7 = 5
Range = 12 - 7 = 5
7
8
9
10
11
12
7
8
9
10
11
12
Interquartile Range


Measure of variation
Also known as midspread


Spread in the middle 50%
Difference between the first and third
quartiles
Data in Ordered Array: 11 12 13 16 16 17
17 18 21
Interquartile Range  Q3  Q1  17.5  12.5  5

Not affected by extreme values
Variance


Important measure of variation
Shows variation about the mean

Sample variance:
n
S2 

Population variance:
2 
 X
i 1
X
i
n 1
N
 X
i 1
i

N
2
2
Standard Deviation



Most important measure of variation
Shows variation about the mean
Has the same units as the original data

Sample standard deviation:
n
S

Population standard deviation:

 X
i 1
X
i
2
n 1
N
 X
i 1
i
N

2
Comparing Standard Deviations
Data A
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
S = 3.338
Data B
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
S = .9258
Data C
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5
S = 4.57
Coefficient of Variation

Measures relative variation

Always in percentage (%)

Shows variation relative to mean


Is used to compare two or more sets of data
measured in different units
S
CV  
X

100%

Comparing Coefficient
of Variation

Stock A:



Stock B:



Average price last year = $50
Standard deviation = $5
Average price last year = $100
Standard deviation = $5
Coefficient of variation:

Stock A:

Stock B:
S
CV  
X

 $5 
100%  
100%  10%

 $50 
S
CV  
X

 $5 
100%  
100%  5%

 $100 
Shape of a Distribution

Describes how data is distributed

Measures of shape

Symmetric or Skewed
(-) Left-Skewed
Mean < Median < Mode
Symmetric
Mean = Median =Mode
(+) Right-Skewed
Mode < Median < Mean
Coefficient of Correlation

Measures the strength of the linear
relationship between two quantitative
variables
n
r
 X
i 1
n
 X
i 1
i
i
 X Yi  Y 
X
2
n
 Y  Y 
i 1
i
2
Features of
Correlation Coefficient

Unit free

Ranges between –1 and 1



The closer to –1, the stronger the negative linear
relationship
The closer to 1, the stronger the positive linear
relationship
The closer to 0, the weaker any positive linear
relationship
Scatter Plots of Data with
Various Correlation Coefficients
Y
Y
Y
X
r = -1
X
r = -.6
Y
X
r=0
Y
r = .6
X
r=1
X
Chapter Summary

Described measures of central tendency

Mean, median, mode, midrange

Discussed quartile

Described measure of variation


Illustrated shape of distribution


Range, interquartile range, variance and standard
deviation, coefficient of variation
Symmetric, Skewed
Discussed correlation coefficient
Stem-n-leaf plot





A class took a test. The students' got the following scores.
94 85 74 85 77 100 85 95 98 95
First let us draw the stem and leaf plot
This makes it easy to see that the mode, the most common score is
an 85 since there are three of those scores and all of the other
scores have frequencies of either one or tow.
It will also make it easier to rank the scores
Stem-n-leaf

This will make it easier to find the median and
the quartiles. There are as many scores above
the median as below. Since there are ten
scores, and ten is an even number, we can
divide the scores into two equal groups with
no scores left over. To find the median, we
divide the scores up into the upper five and
the lower five.
Box Plot








The five-number summary is an abbreviated way to describe a
sample. The five number summary is a list of the following numbers:
Minimum
First (Lower) Quartile,
Median,
Third (Upper) Quartile,
Maximum
The five number summary leads to a graphical representation of a
distribution called the boxplot. Boxplots are ideal for comparing two
nearly-continuous variables. To draw a boxplot (see the example in
the figure below), follow these simple steps:
The ends of the box (hinges) are at the quartiles, so that the length
of the box is the .
Box plots



The median is marked by a line within the box.
The two vertical lines (called whiskers) outside the
box extend to the smallest and largest observations
within of the quartiles.
Observations that fall outside of are called extreme
outliers and are marked, for example, with an open
circle. Observations between and are called mild
outliers and are distinguished by a different mark,
e.g., a closed circle.
Example of boxplot






The Density of Nitrogen - A Comparison of Two Samples
Lord Raleigh was one of the earliest scientists to study the
density of nitrogen. In his studies, he noticed something
peculiar. The density of nitrogen produced from chemical
compounds tended to be smaller than the density of nitrogen
produced from the air
However, he was working with fairly small samples, and the
question is, was he correct in his conjecture?
Lord Raleigh's measurements which first appeared in
Proceedings, Royal Society (London, 55, 1894 pp. 340-344) are
produced below. The units are the mass of nitrogen filling a
certain flask under specified pressure and temperature.
1.
Calculate the summary statistics for each set of data.
2.
Construct side-by-side box plots.
Chemical
Atmospheric
2.30143
2.2989
2.29816
2.30182
2.31017
2.30986
2.3101
2.31001
2.29869
2.2994
2.29849
2.31024
2.3101
2.31028
2.29889
2.30074
2.30054
2.31163
2.30956
Box Plot
Atmospheric pressure has 9 observations and chemical
pressure has 10 values. Use following five points to draw box
plots and compare the two variables
Max 2.30182
2.31163
Min 2.29816
2.30956
Q1
2.29874
2.31001
Median
2.29915
2.3101
Q3
2.30069
2.31024