Business statistics, 7e, by groebner ch03

After completing this chapter, you should be able to:  Compute and interpret the mean, median, and mode for a set of data  Compute the range, variance, and standard deviation and know

After completing this chapter, you should be able to:

 Compute and interpret the mean, median, and mode for a set of data

 Compute the range, variance, and standard deviation and know what these values mean

 Construct and interpret a box and whisker graph

 Compute and explain the coefficient of variation and

z scores

 Use numerical measures along with graphs, charts, and

tables to describe data

Chapter Goals

Chapter Topics

 Measures of Center and Location

 Mean, median, mode

 Other measures of Location

 Weighted mean, percentiles, quartiles

 Measures of Variation

 Range, interquartile range, variance and standard deviation, coefficient of variation

 Using the mean and standard deviation together

 Coefficient of variation, z-scores

Range Percentiles

Interquartile Range Quartiles

Measures of Center and Location

Center and Location

x n

x x

i

i

i W

w

x w w

x

w X

Overview

Mean (Arithmetic Average)

x n

x N

Mean (Arithmetic Average)

 The most common measure of central tendency

 Mean = sum of values divided by the number of values

 Affected by extreme values (outliers)

4 3

Median

 To find the median, sort the n data values from low to high (sorted data is called a

data array )

 Find the value in the i = (1/2)n position

 The i th position is called the Median Index Point

 If i is not an integer, round up to next highest integer

(continued)

Median Example

 Note that n = 13

 Find the i = (1/2)n position:

i = (1/2)(13) = 6.5

 Since 6.5 is not an integer, round up to 7

 The median is the value in the 7th position :

M d = 12

(continued)

Data array:

4, 4, 5, 5, 9, 11, 12, 14, 16, 19, 22, 23, 24

Shape of a Distribution

 Describes how data is distributed

Mean = Median Mean < Median Median < Mean

Right-Skewed Left-Skewed Symmetric

 A measure of location

 The value that occurs most often

 Not affected by extreme values

 Used for either numerical or categorical data

 There 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 = 5

0 1 2 3 4 5 6

No Mode

26

164

2 8 12 4

8) (2

7) (8

6) (12

5)

(4 w

x

w X

i

i i W

 Five houses on a hill by the beach

 Mean is generally used, unless extreme values (outliers) exist

 Then Median is often used, since the median is not sensitive to

extreme values.

 Example: Median home prices may be reported for a region – less sensitive to outliers

Which measure of location

is the “best”?

Other Location Measures

The p th percentile in a data array:

 p% are less than or equal to this

value

 (100 – p)% are greater than or

equal to this value

(where 0 ≤ p ≤ 100)

11.4

(19) 100

60 (n)

100

p

If i is not an integer, round up to the next higher integer value

So use value in the

i = 12 th position

Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22

 Example: Find the first quartile

(n = 9)

Q1 = 25th percentile, so find i : i = (9) = 2.25

so round up and use the value in the 3rd position: Q1 = 13

25 100

Box and Whisker Plot

 A graphical display of data using a central “box” and extended “whiskers”:

Constructing the Box and Whisker Plot

Outliers Lower 1st Median 3rd Upper Limit Quartile Quartile Limit

 The center box extends from Q1 to Q3

 The line within the box is the median

 The whiskers extend to the smallest and largest values within the calculated limits

 Outliers are plotted outside the calculated limits

Shape of Box and Whisker Plots

 The Box and central line are centered between the

endpoints if data is symmetric around the median

 (A Box and Whisker plot can be shown in either

vertical or horizontal format)

Distribution Shape and Box and Whisker Plot

Right-Skewed

Box-and-Whisker Plot Example

 Below is a Box-and-Whisker plot for the following data:

0 2 2 2 3 3 4 5 6 11 27

0 2 3 6 12 27

Min Q 1 Q 2 Q 3 Max

*

Upper limit = Q3 + 1.5 (Q3 – Q1) = 6 + 1.5 (6 – 2) = 12

27 is above the upper limit so is shown as an outlier

Measures of Variation

Variation

Variance Standard Deviation Coefficient of

Variation Population

Variance

Sample Variance

Population Standard Deviation

Sample Standard Deviation

Range

Interquartile

Range

 Measures of variation give information on

the spread or variability of the data values.

Variation

Same center, different variation

 Simplest measure of variation

 Difference between the largest and the smallest observations:

Range = x maximum – x minimum

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Range = 14 - 1 = 13

Example:

 Ignores the way in which data are distributed

 Sensitive to outliers

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

Interquartile Range

 Can eliminate some outlier problems by using

 Eliminate some high-and low-valued

observations and calculate the range from the

remaining values.

 Interquartile range = 3 rd quartile – 1 st quartile

Interquartile Range Example

Median (Q2) X maximum

 Average of squared deviations of values from

N

1 i

2 i

) x

(x s

n

1 i

2 i







Standard Deviation

 Most commonly used measure of variation

 Shows variation about the mean

 Has the same units as the original data

 Population standard deviation:

 Sample standard deviation:

N

μ)

(x σ

N

1 i

2 i

n

2 i

 

1 8

16) (24

16) (14

16) (12

16) (10

1 n

) x (24

) x (14

) x (12

) x

(10 s

2 2

2 2

2 2

2 2

Comparing Standard Deviations

Mean = 15.5

s = 3.338

Same mean, but different standard deviations:

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

100% x

deviation, but stock B is less variable relative

x s

CV B         

 If the data distribution is bell-shaped, then the interval:

 contains about 68% of the values in

the population or the sample

The Empirical Rule

 contains about 95% of the values in

the population or the sample

 contains about 99.7% of the values

in the population or the sample

The Empirical Rule

 Regardless of how the data are distributed,

at least (1 - 1/k 2 ) of the values will fall within

k standard deviations of the mean

 Examples:

(1 - 1/1 2 ) = 0% …… k=1 (μ ± 1σ) (1 - 1/2 2 ) = 75% … k=2 (μ ± 2σ) (1 - 1/3 2 ) = 89% ……… k=3 (μ ± 3σ)

Tchebysheff’s Theorem

within

At least

 A standardized data value refers to the number of standard deviations a value is from the mean

sometimes referred to as z-scores

Standardized Data Values

(number of standard deviations x is from μ)

Standardized Population Values

σ

μ x

(number of standard deviations x is from μ)

Standardized Sample Values

s

x x

 IQ scores in a large population have a

bell-shaped distribution with mean μ = 100 and standard deviation σ = 15

Find the standardized score (z-score) for a person with an IQ of 121

Someone with an IQ of 121 is 1.4 standard deviations above the mean

Standardized Value Example

1.4 15

100

121 σ

μ

x

Answer:

Using Microsoft Excel

 Descriptive Statistics are easy to obtain from Microsoft Excel

Data / data analysis / descriptive statistics

 Enter details in dialog box

Using Excel

 Select:

Data / data analysis / descriptive statistics

 Enter dialog box

Excel output

Microsoft Excel descriptive statistics output,

using the house price data:

House Prices:

$2,000,000 500,000 300,000 100,000 100,000

Chapter Summary

 Described measures of center and location

 Mean, median, mode, weighted mean

 Discussed percentiles and quartiles

 Created Box and Whisker Plots

 Illustrated distribution shapes

 Symmetric, skewed

Chapter Summary

 Described measure of variation

 Range, interquartile range, variance, standard deviation, coefficient of variation

 Discussed Tchebysheff’s Theorem

 Calculated standardized data values

(continued)