Statistics for business decision making and analysis robert stine and foster chapter 04

Chapter 4

Describing Numerical
Data

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4.1 Summaries of Numerical Variables
Can 500 different songs fit on the iPod
Shuffle?


To answer this question we must understand the
typical length of a song and the variation of song
sizes around the typical length



We can do this using summary statistics

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4.1 Summaries of Numerical Variables
A Subset of the Data

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4.1 Summaries of Numerical Variables
The Median


Value in the middle of a sorted list of numerical
values (a typical value)



Half of the values fall below the median; half fall
above



It is the 50th Percentile
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4.1 Summaries of Numerical Variables
Common Percentiles


Lower Quartile = 25th Percentile



Upper Quartile = 75th Percentile



One quarter of the values fall below the lower
quartile and one quarter fall above the upper
quartile
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4.1 Summaries of Numerical Variables
The Interquartile Range (IQR)
IQR = 75th Percentile – 25th Percentile


A measure of variation based on quartiles



Used to accompany the median

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4.1 Summaries of Numerical Variables
The Range
Range = Maximum - Minimum


Maximum Value = 100th Percentile



Minimum Value = 0th Percentile



Another measure of variation; not preferred
because based on extreme values
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4.1 Summaries of Numerical Variables
The Five Number Summary






Minimum
Lower Quartile
Median
Upper Quartile
Maximum

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4.1 Summaries of Numerical Variables
The Five Number Summary for Song Sizes






Minimum = 0.148 MB
Lower Quartile = 2.85 MB
Median = 3.5015 MB
Upper Quartile = 4.32 MB
Maximum = 21.622 MB

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4.1 Summaries of Numerical Variables
Summary Statistics for Song Sizes


Median = 3.5015 MB



IQR = 4.32 MB – 2.85 MB = 1.47 MB



Range = 21.622 MB – 0.148 MB = 21.474 MB

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4.1 Summaries of Numerical Variables
The Mean (Average)


Arithmetic average; divide the sum of the values
by the number of values (another typical value)



The symbol y represents the variable of interest



The symbol

y

read “y bar” represents the mean

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4.1 Summaries of Numerical Variables
The Mean (Average)

y
1  y 2  L  yn
y
n

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4.1 Summaries of Numerical Variables
The Variance (s2)


Is a measure of variation based on the mean



How far a value is from the mean is known as its
deviation; the variance is the average of the squared
deviations

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4.1 Summaries of Numerical Variables

The Variance

s

2

y1  y 



2

  y2  y   L   yn  y 
2

2

n 1

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4.1 Summaries of Numerical Variables
The Standard Deviation (SD)




Is the square root of the variance

s s

2 the original units of the data (the
Is a measure of variability in
variance results in squared units)

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4.1 Summaries of Numerical Variables
Summary Statistics for Song Sizes


Mean = 3.7794 MB



Variance = 2.584 MB²



SD = 1.607 MB

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4M Example 4.1: MAKING M&M’s

Motivation
How many M&M’s are needed to fill a bag
labeled to weigh 1.6 ounces?

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4M Example 4.1: MAKING M&M’s
Method
Data are weights of 72 plain chocolate M&M’s taken
from several packages. To get a measure of the
amount of variation relative to the typical size, we
use the ratio of the standard deviation to the
mean (known as the coefficient of variation).

s
cv 
y
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4M Example 4.1: MAKING M&M’s
Mechanics
Mean Weight = 0.86 gm
SD = 0.04 gm
Cv = 0.04 gm / 0.86 gm = 0.0465

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4M Example 4.1: MAKING M&M’s
Message
Since the SD is quite small compared to the mean
(with a cv of about 5%) the results suggest that 53
pieces are usually enough to fill a bag.
A bag labeled 1.6 ounces weighs about 45.36 grams.
Since there is little variability around the typical weight of
an M&M, we can calculate the number of pieces to fill a
1.6 ounce bag as 45.36/0.86.
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4.2 Histograms and the
Distribution of Numerical Data
Histograms


Plot the distribution of a numerical variable by
showing counts of values occurring within
adjacent intervals



Similar to bar charts but designed for continuous
quantitative data (bar charts are only appropriate
for discrete categories)

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4.2 Histograms and the
Distribution of Numerical Data
Histogram of Song Sizes

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4.2 Histograms and the
Distribution of Numerical Data
Histogram of Song Sizes


Indicates a few very long songs (outliers)



The graph devotes more than half of its area to
show less than 1% of the songs (white space
rule: graphs with mostly white space can be
improved by changing the interval of the plot to
focus on the data rather than the white space)
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4.3 Boxplot
Graph of the Five Number Summary

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