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

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

Chapter 13 Section 5 - Slide 3 Copyright © 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 5 Copyright © 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.

Chapter 13 Section 5 - Slide 6 Copyright © 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.

Chapter 13 Section 5 - Slide 7 Copyright © 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

Chapter 13 Section 5 - Slide 8 Copyright © 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 9 Copyright © 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: 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 = Median = 7.5

Chapter 13 Section 5 - Slide 10 Copyright © 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 11 Copyright © 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.

Chapter 13 Section 5 - Slide 12 Copyright © 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 13 Copyright © 2009 Pearson Education, Inc. Example--dog weights 85, 92, 88, 75, 94, 88, 84, 101 (continued) a. Mean b. Median-rank the data 75, 84, 85, 88, 88, 92, 94, 101 The median is 88.

Chapter 13 Section 5 - Slide 14 Copyright © 2009 Pearson Education, Inc. Example--dog weights 85, 92, 88, 75, 94, 88, 84, 101 c. Mode-the number that occurs most frequently. The mode is 88. d. Midrange = (L + H)/2 = ( )/2 = 88 e. Rank the measures, lowest to highest 88, 88, 88,

Chapter 13 Section 5 - Slide 15 Copyright © 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 16 Copyright © 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 2 nd 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 17 Copyright © 2009 Pearson Education, Inc. To Find the Quartiles of a Set of Data (continued) 3.The first quartile, Q 1, is the median of the lower half of the data; that is, Q 1, is the median of the data less than Q 2. 4.The third quartile, Q 3, is the median of the upper half of the data; that is, Q 3 is the median of the data greater than Q 2.

Chapter 13 Section 5 - Slide 18 Copyright © 2009 Pearson Education, Inc. Example: Quartiles The weekly grocery bills for 23 families are as follows. Determine Q 1, Q 2, and Q

Chapter 13 Section 5 - Slide 19 Copyright © 2009 Pearson Education, Inc. Example: Quartiles (continued) Order the data: Q 2 is the median of the entire data set which is 190. Q 1 is the median of the numbers from 50 to 172 which is 95. Q 3 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 21 Copyright © 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 22 Copyright © 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 23 Copyright © 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.

Chapter 13 Section 5 - Slide 24 Copyright © 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 25 Copyright © 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 26 Copyright © 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.

Chapter 13 Section 5 - Slide 27 Copyright © 2009 Pearson Education, Inc. Example (continued), mean = , , , , ,225  ,756  (  297) 2 = 88,209  (Data  Mean) 2 Data  Mean Data

Chapter 13 Section 5 - Slide 28 Copyright © 2009 Pearson Education, Inc. Example (continued), mean = 514 The standard deviation is $