Chapter 13 Section 5 - Slide 1Copyright © 2009 Pearson Education, Inc.

AND

Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 5 - Slide 2

Chapter 13

Statistics

Chapter 13 Section 5 - Slide 3Copyright © 2009 Pearson Education, Inc.

WHAT YOU WILL LEARN• Mode, median, mean, and

midrange• Percentiles and quartiles• Range and standard deviation

Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 5 - Slide 4

Section 5

Measures of Central Tendency

Chapter 13 Section 5 - Slide 5Copyright © 2009 Pearson Education, Inc.

Definitions

An average is a number that is representative of a group of data.

The arithmetic mean, or simply the mean, is symbolized by , when it is a sample of a population or by the Greek letter mu, , when it is the entire population.

x

Chapter 13 Section 5 - Slide 6Copyright © 2009 Pearson Education, Inc.

Mean

The mean, , is the sum of the data divided by the number of pieces of data. The formula for calculating the mean is

where represents the sum of all the data and n represents the number of pieces of data.

x

x

xn

x

Chapter 13 Section 5 - Slide 7Copyright © 2009 Pearson Education, Inc.

Example-find the mean

Find the mean amount of money parents spent on new school supplies and clothes if 5 parents randomly surveyed replied as follows:

$327 $465 $672 $150 $230

x x

n

$327 $465 $672 $150 $230

5

$1844

5$368.80

Chapter 13 Section 5 - Slide 8Copyright © 2009 Pearson Education, Inc.

Median

The median is the value in the middle of a set of ranked data.

Example: Determine the median of

$327 $465 $672 $150 $230.

Rank the data from smallest to largest.

$150 $230 $327 $465 $672

middle value

(median)

Chapter 13 Section 5 - Slide 9Copyright © 2009 Pearson Education, Inc.

Example: Median (even data)

Determine the median of the following set of data: 8, 15, 9, 3, 4, 7, 11, 12, 6, 4.

Rank the data:

3 4 4 6 7 8 9 11 12 15

There are 10 pieces of data so the median will lie halfway between the two middle pieces (the 7 and 8). The median is (7 + 8)/2 = 7.5

3 4 4 6 9 11 12 157 8

Median = 7.5

Chapter 13 Section 5 - Slide 10Copyright © 2009 Pearson Education, Inc.

Mode

The mode is the piece of data that occurs most frequently.

Example: Determine the mode of the data set: 3, 4, 4, 6, 7, 8, 9, 11, 12, 15.

The mode is 4 since it occurs twice and the other values only occur once.

Chapter 13 Section 5 - Slide 11Copyright © 2009 Pearson Education, Inc.

Midrange

The midrange is the value halfway between the lowest (L) and highest (H) values in a set of data.

Example: Find the midrange of the data set $327, $465, $672, $150, $230.

Midrange =

lowest value + highest value

2

Midrange =

$150 + $672

2$411

Chapter 13 Section 5 - Slide 12Copyright © 2009 Pearson Education, Inc.

Example

The weights of eight Labrador retrievers rounded to the nearest pound are 85, 92, 88, 75, 94, 88, 84, and 101. Determine the

a) mean b) median

c) mode d) midrange

e) rank the measures of central tendency from lowest to highest.

Chapter 13 Section 5 - Slide 13Copyright © 2009 Pearson Education, Inc.

Example--dog weights 85, 92, 88, 75, 94, 88, 84, 101 (continued)

a. Mean

b. Median-rank the data75, 84, 85, 88, 88, 92, 94, 101

The median is 88.

x 85 92 88 75 94 88 84 101

8

707

888.375

Chapter 13 Section 5 - Slide 14Copyright © 2009 Pearson Education, Inc.

Example--dog weights 85, 92, 88, 75, 94, 88, 84, 101c. Mode-the number that occurs most frequently.

The mode is 88.

d. Midrange = (L + H)/2 = (75 + 101)/2 = 88

e. Rank the measures, lowest to highest88, 88, 88, 88.375

Chapter 13 Section 5 - Slide 15Copyright © 2009 Pearson Education, Inc.

Measures of Position

Measures of position are often used to make comparisons.

Two measures of position are percentiles and quartiles.

Chapter 13 Section 5 - Slide 16Copyright © 2009 Pearson Education, Inc.

To Find the Quartiles of a Set of Data

1. Order the data from smallest to largest.

2. Find the median, or 2nd quartile, of the set of data. If there are an odd number of pieces of data, the median is the middle value. If there are an even number of pieces of data, the median will be halfway between the two middle pieces of data.

Chapter 13 Section 5 - Slide 17Copyright © 2009 Pearson Education, Inc.

To Find the Quartiles of a Set of Data (continued)

3. The first quartile, Q1, is the median of the lower half of the data; that is, Q1, is the median of the data less than Q2.

4. The third quartile, Q3, is the median of the upper half of the data; that is, Q3 is the median of the data greater than Q2.

Chapter 13 Section 5 - Slide 18Copyright © 2009 Pearson Education, Inc.

Example: Quartiles

The weekly grocery bills for 23 families are as follows. Determine Q1, Q2, and Q3.

170 210 270 270 280330 80 170 240 270225 225 215 310 5075 160 130 74 8195 172 190

Chapter 13 Section 5 - Slide 19Copyright © 2009 Pearson Education, Inc.

Example: Quartiles (continued)

Order the data: 50 74 75 80 81 95 130160 170 170 172 190 210 215225 225 240 270 270 270 280310 330

Q2 is the median of the entire data set which is 190.

Q1 is the median of the numbers from 50 to 172 which is 95.

Q3 is the median of the numbers from 210 to 330 which is 270.

Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 5 - Slide 20

Section 6

Measures of Dispersion

Chapter 13 Section 5 - Slide 21Copyright © 2009 Pearson Education, Inc.

Measures of Dispersion

Measures of dispersion are used to indicate the spread of the data.

The range is the difference between the highest and lowest values; it indicates the total spread of the data.

Range = highest value – lowest value

Chapter 13 Section 5 - Slide 22Copyright © 2009 Pearson Education, Inc.

Example: Range

Nine different employees were selected and the amount of their salary was recorded. Find the range of the salaries.

$24,000 $32,000 $26,500

$56,000 $48,000 $27,000

$28,500 $34,500 $56,750 Range = $56,750 $24,000 = $32,750

Chapter 13 Section 5 - Slide 23Copyright © 2009 Pearson Education, Inc.

Standard Deviation

The standard deviation measures how much the data differ from the mean. It is symbolized with s when it is calculated for a sample, and with (Greek letter sigma) when it is calculated for a population.

s

x x 2n 1

Chapter 13 Section 5 - Slide 24Copyright © 2009 Pearson Education, Inc.

To Find the Standard Deviation of a Set of Data

1. Find the mean of the set of data.

2. Make a chart having three columns:Data Data Mean (Data Mean)2

3. List the data vertically under the column marked Data.

4. Subtract the mean from each piece of data and place the difference in the Data Mean column.

Chapter 13 Section 5 - Slide 25Copyright © 2009 Pearson Education, Inc.

To Find the Standard Deviation of a Set of Data (continued)5. Square the values obtained in the Data Mean

column and record these values in the (Data Mean)2 column.

6. Determine the sum of the values in the (Data Mean)2 column.

7. Divide the sum obtained in step 6 by n 1, where n is the number of pieces of data.

8. Determine the square root of the number obtained in step 7. This number is the standard deviation of the set of data.

Chapter 13 Section 5 - Slide 26Copyright © 2009 Pearson Education, Inc.

Example

Find the standard deviation of the following prices of selected washing machines:

$280, $217, $665, $684, $939, $299

Find the mean.

6

299939684665217280

n

xx

5146

3084

Chapter 13 Section 5 - Slide 27Copyright © 2009 Pearson Education, Inc.

Example (continued), mean = 514

421,5160

180,625425939

28,900170684

22,801151665

46,225215299

54,756234280

(297)2 = 88,209297217

(Data Mean)2 Data MeanData

Chapter 13 Section 5 - Slide 28Copyright © 2009 Pearson Education, Inc.

Example (continued), mean = 514

The standard deviation is $290.35.

=-

= »

421,516

6 1

421,516290.35

5

s

s