Chapter Three Describing DataMeasures of Location

GoalsWhen you have completed this chapter, you will be able to:1. Calculate the arithmetic mean, median, mode,

wighted mean, and the geometric mean.2. Explain the characteristics, uses, advantages, and disadvantages of each measure of location.3. Identify the position of the arithmetic mean, median, and mode for both symmetric and skewed distributions.

4. Compute and interpret the range, the mean deviation, the variance, and the standard deviation.

5. Explain the characteristics, uses, advantages, and disadvantages of each measure of dispersion.

6. Understand Chebyshev’s theorem and the Normal, or Empirical, Rule as they relate to a set of observations.

7. Compute and interpret quartiles, the interquartile range, and the coefficient of variation.

1-Introduction

This chapter is concerned with two other numerical ways of describing data, namely, measures of central tendency and measures of dispersion, often called variation or the spread.

Parameter A characteristic of a population.Statistic A characteristic of a sample.

2-Population mean

Sum of all the values

in the populationPopulationMean Number of values

in the population

X

N

2-Population Mean

ExampleListed below are 12 automobile companies and the number of patents granted by the United States government to each last year

2-Population Mean

2-Population Mean

511 385 13

122340

19512

The typical number of patents received by an automobile company is 195. Because we considered all the companies receiving patents, this value is a population parameter.

3-The Sample Mean

Sum of all the values

in the sampleSampleMean Number of values

in the sample

XX

n

3-The Sample Mean

ExampleThe Merrill Lynch Global Fund specializes in long-term obligations of foreign countries. We are interested in the interest rate on these obligations. A random sample of six bonds revealed the following.

3-The Sample Mean

Issue Interest Rate

Australian Gov’t Bonds 9.50%

Belgian Gov’t bonds 7.25

Canadian gov’t bonds 6.50

French Gov’t “B-TAN” 4.75

Italian gov’t bonds 12.00

Spanish gov’t bonds 8.30

3-The Sample Mean

9.50 7.25 8.30

648.3

8.056

X

The arithmetic mean interest rate of the sample of the long-term obligations is 8.05 percent.

4-The Properties of the Arithmetic Mean

1 Every set of interval-level and ratio-level data has a mean.

2 All the values are included in computing the mean.3 A set of data has only one mean. The mean is

unique. 4 Deviation from the mean sum to zero 5 Mean is unduly affected by unsusually large and

small values.

0X X

5-Weighted Mean

1 1 2 2

1 2

n nw

n

w X w X w XX

w w w

or w

wXX

w

5-Weighted Mean

Example The Carter Construction Company pays its hourly employees $6.50, $7.50, or $8.50 per hour. There are 26 hourly employees, 14 are paid at rate $6.50, 10 at the $7.50 rate, and 2 at the $8.50 rate. What is the weighted mean hourly rate paid the 26 employees?

14 6.50 10 7.50 2 8.50

14 10 27.038

wX

6-The Median

Median The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to the smallest. Fifty percent of the observations are above the median and fifty percent below the median

Example The number of rooms in the seven hotels in downtown Pittsburgh is

713, 300, 618, 595, 311, 401, and 292. Find the median. SolutionStep1 Arrange the data in order

292, 300, 311, 401, 595, 618, 713Step2 Select the middle value

292, 300, 311, 401, 595, 618, 713

Hence, the median is 401 rooms.

6-The Median

6-The Median

If the number of observations in the data is even the median of the data is the arithmetic mean of the two middle values.

Example: The number of cloudy days for the top ten cloudiest cities is shown. Find the median. 209, 223, 211, 227, 213, 240, 240, 211, 229, 212

6-The Median

Solution:Arrange the data in order209, 211, 211, 212, 213, 223, 227, 229, 240, 240

213 223Median 218

2

6-The Mode

Mode The value of observation that appears most frequently.

Mode is especially useful in describing nominal and ordinal levels of easurement.

Example The following data represent the duration (in days) of U.S. space shuttle voyages for the years 1992 – 1994. Find the mode. 8, 9, 9, 14, 8, 8, 10, 7, 6, 9, 7, 8, 10, 14, 11, 8, 14, 11

6-The Mode

SolutionIt is helpful to arrange the data in order, although it is not necessary6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 14, 14, 14

Since 8-day voyages occurred 5 times, a frequency larger than any other number, the mode for the data set is 8.

Example Find the mode for the number of coal employees per county for ten selected counties in southwestern Pennsylvania.110, 731, 1031, 84, 20, 118, 1162, 1977, 103, 752

SolutionSince each value occurs only once, there is no mode.

6-The Mode

Example Eleven different automobiles were listed at a speed of 15 miles per hour for stopping distances. The data are shown below. Find the mode.

15, 18, 18, 18, 20, 22, 24, 24, 24, 26, 26SolutionSince 18 and 24 both occur 3 times, the modes are 18 and 24. This data set is said to be bimodal.

6-The Mode

6-The Geometric Mean

The geometric mean of a set of n positive numbers is defined as the nth root of the product of n values.

1 2GM nnX X X

6-The Geometric Mean

Another application of the geometric mean is to find an average percent increase over a period of time.

value at end of periodGM= 1

value at beginning of periodn

6-The Geometric Mean

ExampleThe population of a community in 1988 was 2 persons, by 1998 it was 22. What is the average annual rate of percentage increase during the period?

1022

GM 1 1.271 12

0.271

7- The Mean, Median, and Mode of Grouped Data

Arithmetic Mean of Grouped Data

fXX

n

X is the mid-value, or midpoint, of each classf is the frequency in each classn is the total number of frequency

7- The Mean, Median, and Mode of Grouped Data

Class Frequency30 up to 40 440 up to 50 650 up to 60 860 up to 70 1270 up to 80 980 up to 90 7

90 up to 100 4

Example Below is a frequency table showing the distribution of grade points of students. Find the meian.

7- The Mean, Median, and Mode of Grouped Data

Class Freq Midpoint fX30 up to 40 4 35 14040 up to 50 6 45 27050 up to 60 8 55 44060 up to 70 12 65 78070 up to 80 9 75 67580 up to 90 7 85 595

90 up to 100 4 95 3803280fX

328065.6

50X

7- The Mean, Median, and Mode of Grouped Data

Median of Grouped Data

2Median

nCF

L if

L is the lower limit of the class containing the mediann is the total number of frequenciesf is the frequency in the median classCF is the cumulative number of frequencies in all

the classes preceding the class containing the median.

i is the width of the class in which median lies.

7- The Mean, Median, and Mode of Grouped Data

Example The data involving the selling prices of vehicles at Whitner Pontiac are shown in the table below. What is the median selling price for a new vehicle sold by Whitner Pontiac?

7- The Mean, Median, and Mode of Grouped Data

Selling Price($ thousands)

Number Sold ( f ) CF

12 up to 15 8 8

15 up to 18 23 31

18 up to 21 17 48

21 up to 24 18 66

24 up to 27 8 74

27 up to 30 4 78

30 up to 33 1 79

33 up to 36 1 80

Total 80

7- The Mean, Median, and Mode of Grouped Data

2Median

8031

218,000 3,00017

19,588

nCF

L if

7- The Mean, Median, and Mode of Grouped Data

Example A sample of the daily production of transceivers at Scott Electronics was organized into the following distribution. Estimate the median daily production. (Answer: 105.5)

Daily Production Frequency

80 up to 90 5

90 up to 100 9

100 up to 110 20

110 up to 120 8

120 up to 130 6

130 up to 140 2

7- The Mean, Median, and Mode of Grouped Data

For data grouped into a frequency distribution, the mode can be approximated by the midpoint of the class containing the largest number of class frequencies.

8-Selecting an Average for Data in a Frequency Distribution

The mean is influenced more than the median or mode by a few extremely high or low values. If the distribution is highly skewed, the mean would not be a good average to use. The median and mode would be more representative.

THE END!