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

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Describing Numerical

Data

Chapter 4

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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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A Subset of the Data

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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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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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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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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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The Five Number Summary

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The Five Number Summary for Song Sizes

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Summary Statistics for Song Sizes

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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 read “y bar” represents the meany

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The Mean (Average)

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The Variance (s 2)

 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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The Standard Deviation (SD)

 Is the square root of the variance

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

2

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Summary Statistics for Song Sizes

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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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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)

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v

s c

y



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Mechanics

Mean Weight = 0.86 gm

SD = 0.04 gm

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Message

Since the SD is quite small compared to the mean

(with a c v 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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Histogram of Song Sizes

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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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4.3 Boxplot

Combining Boxplots with Histograms

 Boxplots locate the median and quartiles

and highlight outliers

 The median splits the area of the histogram

in half (unlike the mean, it is resistant or

robust to the effects of outliers)

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4.3 Boxplot

Boxplot with Histogram of Song Sizes

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4.4 Shape of a Distribution

Modes

bimodal; three or more is multimodal

height is uniform

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Symmetry and Skewness

 A distribution is symmetric if the two sides

of its histogram are mirror images

 A distribution is skewed if one tail of the

histogram stretches out farther than the

other

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Distribution of Song Sizes

 The mode lies between 3 and 4 MB

 The distribution is right skewed (the right

tail stretches out farther than the left tail)

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4M Example 4.2:

EXECUTIVE COMPENSATION

Message

The distribution of annual salaries of CEO’s

in 2003 is unimodal, nearly symmetric

around the median of $650,000, and right

skewed The average is $697,000 The

largest salary is $4,000,000.

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Bell-Shaped Distributions and Empirical Rule

 A bell-shaped distribution is symmetric and unimodal

 The empirical rule uses the standard

deviation to describe how data with a

bell-shaped distribution cluster around the mean

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The Empirical Rule

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Standardizing

 Converting data to z-scores

 Z- scores measure the distance from the

mean in standard deviations

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y y z

s





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4.5 Epilog

Can 500 different songs fit on the iPod

Shuffle?

Because of variation, not every collection of 500

songs will fit The longest 500 songs won’t fit

However, based on the typical song size, the

amount of variation in song sizes and the shape

of its distribution, we can say that most

collections of 500 songs will fit!

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Best Practices

histograms and summaries such as the mean and standard deviation

with a graph

when preparing a histogram

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Best Practices (Continued)

 Scale your plots to show data, not empty space

 Anticipate what you will see in a histogram

 Label clearly

 Check for gaps

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 Do not ignore the presence of outliers.

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