chapter 9

1Copyright © 2009 Pearson Education, Inc.

Survey of Mathematics – MM150Unit 9 – Statistics

Mr. Scott VanZuiden, Adjunct ProfessorKaplan University

[email protected]

Welcome to seminar!


Chapter 9

Statistics


WHAT YOU WILL LEARN

• Mode, median, mean, and midrange

• Percentiles and quartiles• Range and standard deviation• z-scores and the normal distribution• Correlation and regression


9.1

Measures of Central Tendency


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


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


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


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.


Example: Median (even data)

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


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.


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


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.


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


Measures of Position

Measures of position are often used to make comparisons.

Two measures of position are percentiles and quartiles.


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.


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.


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


Example: Quartiles continued

Order the data: 50 75 74 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.


9.2

Measures of Dispersion


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


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


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


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.


To Find the Standard Deviation of a Set of Data continued5. 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.


Example

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

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

Find the mean.


Example continued, mean = 514

939

684

665

299

280

217

(Data Mean)2 Data MeanData


Example continued, mean = 514

The standard deviation is $


9.3

The Normal Curve


Types of Distributions

Rectangular Distribution J-shaped distribution

Rectangular Distribution

Values

Fre

quen

cy


Types of Distributions continued

Bimodal Skewed to right


Types of Distributions continued

Skewed to left Normal


Properties of a Normal Distribution

The graph of a normal distribution is called the normal curve.

The normal curve is bell shaped and symmetric about the mean.

In a normal distribution, the mean, median, and mode all have the same value and all occur at the center of the distribution.


Empirical Rule

Approximately 68% of all the data lie within one standard deviation of the mean (in both directions).

Approximately 95% of all the data lie within two standard deviations of the mean (in both directions).

Approximately 99.7% of all the data lie within three standard deviations of the mean (in both directions).


z-Scores

z-scores determine how far, in terms of standard deviations, a given score is from the mean of the distribution.

z

value of piece of data mean

standard deviation

x


Example: z-scores

A normal distribution has a mean of 50 and a standard deviation of 5. Find z-scores for the following values.

a) 55 b) 60 c) 43


Example: z-scores continued


To Find the Percent of Data Between any Two Values

1. Draw a diagram of the normal curve, indicating the area or percent to be determined.

2. Use the formula to convert the given values to z-scores. Indicate these z-scores on the diagram.

3. Look up the percent that corresponds to each z-score in Table 13.7.


To Find the Percent of Data Between any Two Values continued 4.

a) When finding the percent of data between two z-scores on opposite sides of the mean (when one z-score is positive and the other is negative), you find the sum of the individual percents.

b) When finding the percent of data between two z-scores on the same side of the mean (when both z-scores are positive or both are negative), subtract the smaller percent from the larger percent.


To Find the Percent of Data Between any Two Values continued

c) When finding the percent of data to the right of a positive z-score or to the left of a negative z-score, subtract the percent of data between 0 and z from 50%.

d) When finding the percent of data to the left of a positive z-score or to the right of a negative z-score, add the percent of data between 0 and z to 50%.


Example

Assume that the waiting times for customers at a popular restaurant before being seated for lunch are normally distributed with a mean of 12 minutes and a standard deviation of 3 min.

a) Find the percent of customers who wait for at least 12 minutes before being seated.

b) Find the percent of customers who wait between 9 and 18 minutes before being seated.

c) Find the percent of customers who wait at least 17 minutes before being seated.

d) Find the percent of customers who wait less than 8 minutes before being seated.


Solution

a. wait for at least 12 minutes

b. between 9 and 18 minutes


Solution continued

c. at least 17 min d. less than 8 min


9.4

Linear Correlation and Regression


Linear Correlation

Linear correlation is used to determine whether there is a relationship between two quantities and, if so, how strong the relationship is.


Linear Correlation

The linear correlation coefficient, r, is a unitless measure that describes the strength of the linear relationship between two variables. If the value is positive, as one variable

increases, the other increases. If the value is negative, as one variable

increases, the other decreases. The variable, r, will always be a value

between –1 and 1 inclusive.


Scatter Diagrams

A visual aid used with correlation is the scatter diagram, a plot of points (bivariate data). The independent variable, x, generally is a

quantity that can be controlled. The dependent variable, y, is the other

variable. The value of r is a measure of how far a set of

points varies from a straight line. The greater the spread, the weaker the

correlation and the closer the r value is to 0. The smaller the spread, the stronger the

correlation and the closer the r value is to 1.


Correlation


Linear Correlation Coefficient

The formula to calculate the correlation coefficient (r) is as follows:

2 22 2

n xy x yr

n x x n y y


There are five applicants applying for a job as a medical transcriptionist. The following shows the results of the applicants when asked to type a chart. Determine the correlation coefficient between the words per minute typed and the number of mistakes.

Example: Words Per Minute versus Mistakes

934Nancy1041Kendra1253Phillip1167George824Ellen

MistakesWords per MinuteApplicant


We will call the words typed per minute, x, and the mistakes, y.

List the values of x and y and calculate the necessary sums.

Solution

306811156934xy = 2,281y2 = 510x2 =10,711y = 50x = 219

1012118y

Mistakesxyy2 x2x

41536724

WPM

41010016816361442809737121448919264576


Solution continued

The n in the formula represents the number of pieces of data. Here n = 5.

r n xy x y

n x2 x 2 n y 2 y 2

r 5 2281 219 50

5 10,711 219 2 5 510 50 2


Solution continued

11,405 10,950

5 10,711 47,961 5 510 2500

455

53,555 47,961 2550 2500

455

5594 500.86


Solution continued

Since 0.86 is fairly close to 1, there is a fairly strong positive correlation.

This result implies that the more words typed per minute, the more mistakes made.


Linear Regression

Linear regression is the process of determining the linear relationship between two variables.

The line of best fit (regression line or the least squares line) is the line such that the sum of the squares of the vertical distances from the line to the data points (on a scatter diagram) is a minimum.


The Line of Best Fit

Equation:

y mx b, where

m n xy x y

n x2 x 2, and b

y m x n


Example

Use the data in the previous example to find the equation of the line that relates the number of words per minute and the number of mistakes made while typing a chart.

Graph the equation of the line of best fit on a scatter diagram that illustrates the set of bivariate points.


Solution

From the previous results, we know that

m n xy x y

n x2 x 2

m 5(2,281) (219)(50)

5(10,711) 2192

m 455

5594m 0.081


Solution

Now we find the y-intercept, b.

Therefore the line of best fit is y = 0.081x + 6.452

b y m x

n

b 50 0.081 219

5

b 32.261

56.452


Solution continued

To graph y = 0.081x + 6.452, plot at least two points and draw the graph.

8.88230

8.07220

7.26210

yx


Solution continued

chapter 9

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