Chapter 3:Chapter 3:Descriptive MeasuresDescriptive Measures

STP 226: Elements of StatisticsSTP 226: Elements of Statistics

Jenifer BoshesJenifer Boshes

Arizona State UniversityArizona State University

3.1: Measures of Center3.1: Measures of Center

MeanMeanThe The meanmean of a data set is the sum of the of a data set is the sum of the observations divided by the number of observations divided by the number of

observations. (average)observations. (average)

Example 1:Example 1:The following data set is The following data set is

comprised of a set of comprised of a set of homework grades. Find the homework grades. Find the mean homework grade.mean homework grade.

93 87 90 90 82 85 93 87 90 90 82 85

88 90 93 83 9088 90 93 83 90

Interpret:

Example 2:Example 2:The following data set is The following data set is

comprised of the lengths of a comprised of the lengths of a rare orchid (in inches). Find rare orchid (in inches). Find the mean orchid length.the mean orchid length.

13 18 14.5 13 18 14.5

14 15 1414 15 14

Interpret:

MedianMedianTo find the To find the medianmedian of a data set: of a data set:

Arrange the data in increasing order.Arrange the data in increasing order.

• If the number of observations is odd, then the median is the observation If the number of observations is odd, then the median is the observation exactly in the middle.exactly in the middle.

• If the number of observations is even, the median is the mean of the two If the number of observations is even, the median is the mean of the two middle observations in the ordered list.middle observations in the ordered list.

Example 3:Example 3:Find the median homework score.

Example 4:Example 4:Find the median orchid length.

Interpret: Interpret:

93 87 90 90 82 85 93 87 90 90 82 85

88 90 93 83 9088 90 93 83 90

13 18 14.5 13 18 14.5

14 15 1414 15 14

ModeModeThe The modemode of a data set is value that occurs with greatest frequency. of a data set is value that occurs with greatest frequency.

First, find the frequency of each value in the data set.First, find the frequency of each value in the data set.

• If no value occurs more than once, there is no mode.If no value occurs more than once, there is no mode.

• Otherwise, any value that occurs with greatest frequency is a mode.Otherwise, any value that occurs with greatest frequency is a mode.

Example 5:Example 5:Find the mode homework score.

Example 6:Example 6:Find the mode orchid length.

Interpret: Interpret:

93 87 90 90 82 85 93 87 90 90 82 85

88 90 93 83 9088 90 93 83 90

13 18 14.5 13 18 14.5

14 15 1414 15 14

Example 7:Example 7:

Find the mean, median, and mode of each of the data sets.Find the mean, median, and mode of each of the data sets.

4545 9090 7878 8888

6868 1616 8686 4949

8888 8686 8282 7676

Data Set IData Set I

6161 9999 8282

7272 8080 9696

7878 7777 6666

Data Set IIData Set II

Skewed vs. SymmetricSkewed vs. Symmetric

(a)(a) Right skewedRight skewed:: The mean is to the right The mean is to the right of the median. of the median.

(b)(b) SymmetricSymmetric:: The mean is equal to the The mean is equal to the median.median.

(c)(c) Left skewedLeft skewed:: The mean is to the left of The mean is to the left of the median.the median.

When to use each…When to use each…

MedianMedian:: Use the median when your data set Use the median when your data set has very extreme values.has very extreme values.

AA resistant measure resistant measure (or robust)(or robust) is not is not sensitive to the influence of a few extreme sensitive to the influence of a few extreme observations.observations.

ModeMode:: Use the mode when you have qualitative Use the mode when you have qualitative data.data.

Sample MeanSample Mean

Example 8:Example 8:

The exam scores for a student are: 61, The exam scores for a student are: 61, 97, 78, 86, and 73.97, 78, 86, and 73.

(a)(a) Use mathematical notation to Use mathematical notation to represent the individual exam scores.represent the individual exam scores.

(b)(b) Use summation notation to express Use summation notation to express the sum of the five exam scores.the sum of the five exam scores.

(c)(c) Find for the exam data.Find for the exam data.x

3.2: Measures of 3.2: Measures of VariationVariation

Example 1:Example 1:

The exam scores for student A are: 100, The exam scores for student A are: 100, 100, 90, 90, and 70. The exam scores 100, 90, 90, and 70. The exam scores for student B are: 90, 88, 88, 93, and for student B are: 90, 88, 88, 93, and 91. Compare the means and 91. Compare the means and medians.medians.

Who is the better student? Who is more consistent?

RangeRange

Standard Deviation Standard Deviation

The The standard deviationstandard deviation measures measures variation by indicating, on average, variation by indicating, on average, how far the observations are from the how far the observations are from the mean.mean.

Sample Standard Deviation Sample Standard Deviation

1.1. For each observation, calculate the deviation from For each observation, calculate the deviation from the mean.the mean.

2.2. Square this value.Square this value.3.3. Add up the squares.Add up the squares.4.4. Divide by Divide by nn – 1. – 1.5.5. Take the square root.Take the square root.

Example 2:Example 2:

Find the standard deviation for student A:Find the standard deviation for student A:100, 100, 90, 90, and 70. 100, 100, 90, 90, and 70.

1. For each observation, calculate the deviation from the mean.

2. Square this value.3. Add up the squares.4. Divide by n – 1.5. Take the square root.

Example 3:Example 3:

Find the standard deviation for student B:Find the standard deviation for student B: 90, 88, 88, 93, and 91. 90, 88, 88, 93, and 91.

1. For each observation, calculate the deviation from the mean.

2. Square this value.3. Add up the squares.4. Divide by n – 1.5. Take the square root.

What can we say about the relative performance between students A and B?

Comments on Standard DeviationComments on Standard Deviation

ss22 is called the is called the sample variancesample variance..

The units of The units of ss22 are the square of the original units.are the square of the original units.

The units of The units of ss are the same as the original units. are the same as the original units.

ss is ALWAYS ≥ 0. Why? is ALWAYS ≥ 0. Why?

ss is a measure of how much each point deviates from the is a measure of how much each point deviates from the mean deviation.mean deviation.

Do not perform any rounding until the computation is Do not perform any rounding until the computation is complete; otherwise, substantial roundoff error can result.complete; otherwise, substantial roundoff error can result.

Almost all the observations in Almost all the observations in anyany data set lie within three data set lie within three standard deviations to either side of the mean. This is standard deviations to either side of the mean. This is known as known as Chebyshev’s RuleChebyshev’s Rule. .

Example 3:Example 3:• How many observations for student B are within one standard deviation of the mean? • How many observations for student B are within two standard deviation of the mean? • How many observations for student B are within three standard deviation of the mean?

3.3: The Five-Number 3.3: The Five-Number Summary;Summary;

BoxplotsBoxplots

Recall: What does it mean for a statistic to Recall: What does it mean for a statistic to be robust? be robust?

Name a statistic that is not robust.Name a statistic that is not robust.

Name a statistic that is robustName a statistic that is robust

Robustness

QuartilesQuartiles

QuartilesQuartiles divide a data set divide a data set into quarters. into quarters. QQ11, , QQ22, and , and QQ33 are the three quartiles.are the three quartiles.

TheThe second quartilesecond quartile ( (QQ22) ) is is the median of the entire data the median of the entire data set.set.

TheThe first quartilefirst quartile ( (QQ11) ) is the is the median of the portion of the median of the portion of the data set that lies at or below data set that lies at or below QQ22..

TheThe third quartilethird quartile ( (QQ33) ) is the is the median of the portion of the median of the portion of the data set that lies at or above data set that lies at or above QQ22..

Example 1:Example 1:Fifteen people were asked how Fifteen people were asked how

many baseball games they many baseball games they had attended the previous had attended the previous season.season.

Find the quartiles.Find the quartiles.

1212 2525 88 66 11

00 4242 1919 1717 00

6363 1414 2222 3131 3434

1. Order the data.2. Find the median of the data

set. This is Q2.3. Find the median of the data

that lies at or below the median of the entire data set. This is Q1.

4. Find the median of the data that lies at or above the median of the entire data set. This is Q3.

Interquartile Range (IQR)Interquartile Range (IQR)

The The IQRIQR is the difference between the first is the difference between the first and third quartiles; that is, and third quartiles; that is, IQRIQR = = QQ33 – – QQ11..

It is the preferred measure of variation It is the preferred measure of variation when the median is used as the measure when the median is used as the measure of center. Like the median, the of center. Like the median, the IQRIQR is a is a resistant or robust measureresistant or robust measure. .

Example 2:Example 2:

What is the IQR for the baseball data?What is the IQR for the baseball data?

Interpret:Interpret:

Five-Number SummaryFive-Number Summary

MinMin

QQ11

QQ22

QQ33

MaxMax

Example 3:Example 3:

Find the five-number summary for the Find the five-number summary for the baseball data.baseball data.

OutliersOutliers

OutliersOutliers are observations that fall well are observations that fall well outside the overall pattern of the data.outside the overall pattern of the data.

They may result from a recording error, They may result from a recording error, obtaining an observation from a different obtaining an observation from a different population, or an unusual extreme value.population, or an unusual extreme value.

Lower and Upper LimitsLower and Upper Limits

Lower limitLower limit:: QQ11 – 1.5 – 1.5 ·· IQRIQR

Upper limitUpper limit:: QQ33 + 1.5 + 1.5 ·· IQRIQR

Observations that lie outside the upper Observations that lie outside the upper and lower limits – either below the lower and lower limits – either below the lower limit or above the upper limit – are limit or above the upper limit – are potential outliers.potential outliers.

Example 4:Example 4:For the baseball data:For the baseball data:(a)(a) Obtain the lower and upper limits.Obtain the lower and upper limits.

(b)(b) Determine the potential outliers, if any.Determine the potential outliers, if any.

(c)(c) Construct a modified boxplot.Construct a modified boxplot.

• Adjacent values of a set are the most extreme observations that are not potential outliers.

1212 2525 88 66 11

00 4242 1919 1717 00

6363 1414 2222 3131 3434

Steps for Constructing a Modified Steps for Constructing a Modified BoxplotBoxplot

Steps for Constructing a BoxplotSteps for Constructing a Boxplot

BoxplotsBoxplots

Boxplots are useful for comparing two or Boxplots are useful for comparing two or more data sets.more data sets.

Notice how box width and whisker length Notice how box width and whisker length relate to skewness and symmetry.relate to skewness and symmetry.

BibliographyBibliography

Some of the textbook images embedded in Some of the textbook images embedded in the slides were taken from:the slides were taken from:

Elementary StatisticsElementary Statistics, Sixth Edition; by , Sixth Edition; by Weiss; Addison Wesley Publishing Weiss; Addison Wesley Publishing Company Company

Copyright © 2005, Pearson Education, Inc.Copyright © 2005, Pearson Education, Inc.