Essentials of Business Statistics: Communicating with Numbers By Sanjiv Jaggia and Alison Kelly Copyright © 2014 by McGraw-Hill Higher Education. All rights

Essentials of Business Statistics: Communicating with Numbers

By Sanjiv Jaggia and Alison KellyCopyright © 2014 by McGraw-Hill Higher Education. All rights reserved.

3-2Numerical Descriptive Measures

Chapter 3 Learning Objectives

LO 3.1 Calculate and interpret the mean, the median, and the mode.

LO 3.2 Calculate and interpret percentiles and a box plot.LO 3.3 Calculate and interpret the range, the mean absolute

deviation, the variance, the standard deviation, and the coefficient of variation.

LO 3.4 Explain mean-variance analysis and the Sharpe ratio.LO 3.5 Apply Chebyshev’s Theorem, the empirical rule, and z-

scores.LO 3.6 Calculate the mean and the variance for grouped data.LO 3.7 Calculate and interpret the covariance and the correlation

coefficient.


The arithmetic mean is a primary measure of central location.

LO 3.1 Calculate and interpret the arithmetic mean, the median, and the mode.

3.1 Measures of Central Location

x

Sample Mean Population Mean


The mean is sensitive to outliers. Consider the salaries of employees at Acetech.

LO 3.1 3.1 Measures of Central Location

This mean does not reflect the typical salary!


The median is another measure of central location that is not affected by outliers.

When the data are arranged in ascending order, the median is

the middle value if the number of observations is odd, or

the average of the two middle values if the number of observations is even.



The mode is another measure of central location. The most frequently occurring value in a data set Used to summarize qualitative data A data set can have no mode, one mode (unimodal),

or many modes (multimodal). Consider the salary of employees at Acetech

The mode is $40,000 since this value appears most often.



Weighted Mean

Let w1, w2, . . . , wn denote the weights of the sample observations x1, x2, . . . , xn such that

w1 + w2 + . . . + wn = 1, then

i ix w x



In general, the pth percentile divides a data set into two parts: Approximately p percent of the observations have

values less than the pth percentile; Approximately (100 p ) percent of the

observations have values greater than the pth percentile.

LO 3.2 Calculate and interpret percentiles and a box plot.

3.2 Percentiles and Box Plots


Calculating the pth percentile: First arrange the data in ascending order. Locate the position, Lp, of the pth percentile by using

the formula:

We use this position to find the percentile as shown next.

1100p

pL n

LO 3.2 Percentiles and Box Plots


Calculating the pth percentile

Once you find Lp, observe whether or not it is an integer. If Lp is an integer, then the Lpth observation in the

sorted data set is the pth percentile. If Lp is not an integer, then interpolate between two

corresponding observations to approximate the pth percentile.



Both L25 = 2.75 and L75 = 8.25 are not integers, thus

The 25th percentile is located 75% of the distance between the second and third observations, and it is

The 75th percentile is located 25% of the distance between the eighth and ninth observations, and it is

7.34 0.75(8.09 ( 7.34)) 4.23

43.79 0.25(59.45 43.79) 47.71


3-12

A box plot allows you to: Graphically display the distribution of a data set. Compare two or more distributions. Identify outliers in a data set.

Numerical Descriptive Measures

Box

* *

WhiskersOutliers



The box plot displays 5 summary values: Min = smallest value Max = largest value Q1 = first quartile = 25th percentile Q2 = median = second quartile = 50th percentile Q3 = third quartile = 75th percentile


Min Max

3-14

Detecting outliers Calculate IQR, then multiply by 1.5 There are outliers if

Q1 – Smallest Value > 1.5*IQR Largest Value – Q3 > 1.5*IQR

Numerical Descriptive Measures



Measures of dispersion gauge the variability of a data set.

Measures of dispersion include: Range Mean Absolute Deviation (MAD) Variance and Standard Deviation Coefficient of Variation (CV)

LO 3.4 Calculate and interpret the range, the mean absolute deviation, the variance, the standard deviation, and the coefficient of variation.

3.3 Measures of Dispersion


Range

It is the simplest measure. It is focused on extreme values.

Range Maximum Value Minimum Value

LO 3.4 3.3 Measures of Dispersion


Mean Absolute Deviation (MAD) MAD is an average of the absolute difference of each

observation from the mean.

Sample MAD ix x

n

Population MAD ix

N



Variance and standard deviation

For a given sample,

For a given population,

2

2 2 and 1

ix xs s s

n

2

2 2 and ix

N



Coefficient of variation (CV) CV adjusts for differences in the magnitudes of the

means. CV is unitless, allowing easy comparisons of mean-

adjusted dispersion across different data sets.

Sample CV

Population CV

s

x



Mean-variance analysis: The performance of an asset is measured by its rate of

return. The rate of return may be evaluated in terms of its reward

(mean) and risk (variance). Higher average returns are often associated with higher

risk. The Sharpe ratio uses the mean and variance to

evaluate risk.

LO 3.5 Explain mean-variance analysis and the Sharpe Ratio.

3.4 Mean-Variance Analysis andthe Sharpe ratio


Sharpe Ratio Measures the extra reward per unit of risk. For an investment І , the Sharpe ratio is computed as:

where is the mean return for the investment is the mean return for a risk-free asset is the standard deviation for the investment

Sharpe Ratiox R

s

LO 3.4 3.4 Mean-Variance Analysis andthe Sharpe Ratio


Chebyshev’s Theorem For any data set, the proportion of observations that lie

within k standard deviations from the mean is at least 11/k2 , where k is any number greater than 1.

Consider a large lecture class with 280 students. The mean score on an exam is 74 with a standard deviation of 8. At least how many students scored within 58 and 90?With k = 2, we have 1(1/2)2 = 0.75. At least 75% of 280 or 210 students scored within 58 and 90.

LO 3.5 Apply Chebyshev’s Theorem and the empirical rule.

3.5 Analysis of Relative Location


The Empirical Rule: Approximately 68% of all observations fall in the interval

Approximately 95% of all

observations fall in the interval

Almost all observations fall in the interval

x s

2x s

3x s

LO 3.5 Analysis of Relative Location


LO 3.5 Analysis of Relative Location z-Scores

Often useful to use z-score to denote relative location A sample value’s z-score indicates that sample value’s

distance from the mean z-score calculated as

A z-score of 1.25 indicates that that sample value is 1.25 standard deviations above the mean. A z-score of –2.43 would indicate a sample value that is 2.43 standard deviations below the mean.

s

xxz


When data are grouped or aggregated, we use these formulas:

where mi and i are the midpoint and frequency of the ith class, respectively.

LO 3.6 Calculate the mean and the variance for grouped data.

2

2

2

Mean:

Variance: 1

Standard Deviation:

i i

i i

mx

n

m xs

n

s s

3.6 Summarizing Grouped Data


The covariance (sxy or xy) describes the direction of the linear relationship between two variables, x and y.

The correlation coefficient (rxy or xy) describes both the direction and strength of the relationship between x and y.

LO 3.7 Calculate and interpret the covariance and the correlation coefficient.

3.7 Covariance and Correlation


The sample covariance sxy is computed as

1

i ixy

x x y ys

n

The population covariance xy is computed as

i x i y

xy

x y

N

LO 3.7 Covariance and Correlation


The sample correlation rxy is computed as

xyxy

x y

sr

s s

The population correlation xy is computed as

Note, 1 < rxy < +1 or 1 < xy < +1

xyxy

x y

LO 3.7 Covariance and Correlation

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Essentials of Business Statistics: Communicating with Numbers By Sanjiv Jaggia and Alison Kelly Copyright © 2014 by McGraw-Hill Higher Education. All rights