Overview

OverviewOverview

3.1 Measures of Center

3.2 Measures of Variability

3.4 Measures of Position

3.6 Robust Measures

3.1 Measures of Center3.1 Measures of CenterObjectives:By the end of this section, I will beable to…

1)Calculate the mean for a given data set.

2)Find the median, and describe why the median is sometimes preferable to the mean.

3)Find the mode of a data set.

4)Describe how skewness affects these measures of center.

The MeanThe Mean

Most well known and widely used measure of center

Simply add up all the numbers and divide by how many numbers you have.

NotationNotation

Statisticians like to use specialized notation.

Sample size - how many observations you have in your sample data set, is always denoted by n

ith data value by xi, where i is simply an index or counter indicating a data point

“add them together” is Σ (capital sigma)

The sample mean is called (pronounced “x-bar”)

X

The sample meanThe sample mean

Written as

In plain English, this just means that, in orderto find the mean x, we

1. Add up all the data values, giving us Σx.

2. And divide by how many observations are in the data set, giving us Σx /n.

xx

n

The Population Mean The Population Mean Mean value of a population is usually unknown

Use x to estimate

Denote the population mean with (mu)

Population size is denoted by N.

The mean is sensitive to the presence of extreme values

x

N

The MedianThe Median The middle data value when the data are put into

ascending order

Half of the data values lie below the median and half lie above

If the sample size n is odd, then the median is a unique middle value.

That is, observation when the data are put in ascending order.

If the sample size n is even, then the median is the mean of the two data values in the middle.

That is, the median is the mean of the two data values that lie on either side of the position.

1

2

thn

1

2

thn

The ModeThe Mode

French speakers will recognize that the term mode in French refers to fashion

The popularity of clothing often depends on just which style is in fashion

In a data set, the value that is most “in fashion” is the value that occurs the most

The mode of a data set is the data value that occurs with the greatest frequency

Example 3.5 - Cost of Example 3.5 - Cost of mathematical journalsmathematical journalsThe rising cost of research journals has been taking an increasing bite out of libraryand research budgets. Table 3.3 contains the annual subscription cost of ten researchjournals in mathematics and statistics for 2006. Find the following.

a. The mean journal subscription costb. The median journal subscription costc. The mode journal subscription cost

Example 3.5 continuedExample 3.5 continued

Table 3.3 Annual subscription cost for ten research journals

Example 3.5 continued Example 3.5 continued

Solution a.The sample mean journal cost is

250 250 402 467 850 1022 1582 1653 1744 3631

1011,851

10$1185.10

Xx

n

Example 3.5 continuedExample 3.5 continued

b.

Since we have n 10 journals, the median is the mean of the two data valuesthat lie on either side of the

The median is the mean of the 5th and 6th

data values, $850 and $1022

median journal cost =

1 10 15.5

2 2

th ththnposition

850 1022$936

2

Example 3.5 continuedExample 3.5 continued

c.

The mode is the data value that occurs with

the greatest frequency.

Only two journals that cost $250 each.

No other cost occurs more than once.

Mode = $250.

Mode is not a very good measure of center for this data set because it is the minimum value.

Illustrates a weakness of using the mode as a measure of center.

How Skewness Affects the How Skewness Affects the Mean and MedianMean and MedianFor a right-skewed distribution, the mean is

larger than the median.

For a left-skewed distribution, the median is larger than the mean.

For a symmetric unimodal distribution, the mean, median, and mode are fairly close to one another.

FIGURE 3.5 How skewness affects the mean and median.

Exploratory Data AnalysisExploratory Data Analysis

Using graphical methods to compare numerical statistics

FIGURE 3.6 Dotplots of the percentage net price change for the Dow Jones Industrial Average, the randomly selected darts portfolio, and the professionally selected portfolio.

SummarySummary

The sample mean represents the sum of the data values in the sample divided by the sample size (n).

The population mean represents the sum of the data values in the population divided by the population size (N).

The mean is sensitive to the presence of extreme values.

x

SummarySummary

The median occupies the middle position when the data are put in ascending order and is not sensitive to extreme values.

The mode is the data value that occurs with the greatest frequency.

Modes can be applied to categorical data as well as numerical data but are not always reliable as measures of center.

SummarySummary

The skewness of a distribution can often tell us something about the relative values of the mean and the median.

3.2 Measures of Variability3.2 Measures of Variability

Objectives:By the end of this section, I will beable to…

1)Understand and calculate the range of a data set.

2)Explain in my own words what a deviation is.

3)Calculate the variance and the standard deviation for a population or a sample.

The RangeThe Range

The difference between the largest value and the smallest value in the data set:

range = largest value – smallest value

Simplest measure of variability

Larger range is an indication of greater variability

Example 3.8 - Range of the Example 3.8 - Range of the volleyball teams’ heightsvolleyball teams’ heights

Calculate the range of player heights for each of the WMU and NCU teams.

FIGURE 3.11 Comparative dotplots of the heights of two volleyball teams.

Example 3.8 continuedExample 3.8 continuedSolution

From Figure 3.11shows WMU heights are more spread out than NCU heights.

Range of WMU team should be larger than the range of the NCU team, reflecting greater variability.

rangeWMU = 75 - 60 = 15 inches

rangeNCU = 72 – 66 = 6 inches

What Is a Deviation?What Is a Deviation?

A deviation for a given data value x is the difference between the data value and the mean of the data set.

For a sample, the deviation equals x - x.

For a population, the deviation equals x - .

Data value is larger than the mean, the deviation will be positive

DeviationDeviation Data value is smaller than the mean, the

deviation will be negative

Data value equals the mean, the deviation will be zero

Deviation can roughly be thought of as the distance between a data value and the mean

The deviation can be negative while distance is always positive

Deviation not useful measure of spread because sum of deviations is always zero.

Population Variance Population Variance

Symbolized by the lowercase Greek letter sigma squared,

Is the mean of the squared deviations in the population and is found by

22 x

N

The Population Standard The Population Standard Deviation Deviation

The square root of the variance

Represents a distance from the mean that is representative for that data set

Not the mean deviation, which is always zero

2x

N

Sample Variance Sample Variance ss22

Based on the idea of finding the sum of the squared deviations x – x)2 and then dividing by the sample size to get the mean squared deviation

Statisticians found a better estimate by dividing by n - 1

22

1

x xs

n

Sample Standard Deviation Sample Standard Deviation ss

The square root of the sample variance s2

Second most important statistic

The value of s may be interpreted as the typical difference between a data value and the sample mean

22

1

x xs s

n

Computational FormulasComputational Formulas

Population variance: Population standard deviation:

22

2

xx

NN

22

2

xx

NN

Sample variance: Sample standard deviation:

22

2

1

xx

nsn

22

2

1

xx

nsn

Example 3.15 - Calculating the Example 3.15 - Calculating the population variance and population variance and population standard deviation population standard deviation using the calculator.using the calculator.

Table 3.13 lists the amount of farmland (in 1000s of acres) in each county in the stateof Connecticut. Since the data set contains all N = 8 counties in Connecticut, it canbe considered a population. Calculate the population variance and population standarddeviation using the calculator.

Example 3.15 continuedExample 3.15 continued

Table 3.13 Farmland in Connecticut

Example 3.15 continuedExample 3.15 continued

The population standard deviation is therefore:

The standard deviation of farmland for all counties in Connecticut is almost 25,100 acres.

2 629.9998438 25.1

SummarySummary

The simplest measure of variability, or measure of spread, is the range.

The range is simply the difference between the maximum and minimum values in a data set

The range has drawbacks because it relies on the two most extreme data values.

A deviation is the difference between a data value and the mean of the data values.

SummarySummary

The variance and standard deviation are measures of spread that utilize all available data values.

The population variance can be thought of as the mean squared deviation.

The standard deviation is the square root of the variance.

Standard deviation is a typical deviation, that is, the typical difference between a data value and the mean.

3.4 Measures of Position3.4 Measures of PositionObjectives:By the end of this section, I will beable to…

1)Find percentiles for both small and large data sets.

PercentilePercentile

Let p be any integer between 0 and 100.

The pth percentile of a data set is the data value at which p percent of the values in the data set are less than or equal to this value.

Example 3.24 - Finding Example 3.24 - Finding percentiles of a small data setpercentiles of a small data setYolanda would like to go to a prestigious graduate school of the arts. She knows thatthis school accepts only those students who score at the 75th percentile or higher ina grueling dance audition. The following data represent the dance audition scores ofYolanda’s group. Yolanda scored 85. Find the 75th percentile of the data set. Will Yolanda be accepted at the prestigious graduate school of the arts?

78 56 89 44 65 94 81 62 75 85 30 68

Example 3.24 continuedExample 3.24 continued

SolutionStep 1: Sort the data into ascending order30 44 56 62 65 68 75 78 81 85 89 94

Step 2:Since we want the 75th percentile, p=75.There are 12 scores, so n=12. Calculate

So, i = 9.75

12 9100 100

pi n

Example 3.24 continuedExample 3.24 continuedStep 3:

Here, since i is an integer, the 75th percentile

is the mean of the data values in positions 9

and 10.

Data value in the ninth position is 81. Data value in the tenth position is 85. Mean of these values is 83. Thus, the 75th

percentile is 83.

Yolanda’s dance score of 85 is therefore

above the 75th percentile. She will be

accepted to the prestigious graduate school.

OutliersOutliers

Extremely large or extremely small data value relative to the rest of the data set

May represent a data entry error, or it may be genuine data

Farther than three standard deviations from the mean

SummarySummary

Measures of position, which tell us the position that a particular data value holds relative to the rest of the data set.

The pth percentile of a dataset is the value at which p percent of the values in the data set are less than or equal to this value.

3.6 Robust Measures3.6 Robust Measures

Objectives:By the end of this section, I will beable to…

1)Find quartiles and the interquartile range.

2)Calculate the five-number summary of a data set.

3)Construct a boxplot for a given data set.

4)Apply robust detection of outliers.

QuartilesQuartiles

Divide the data set into quarters

FIGURE 3.31

QuartilesQuartilesEach part contains 25% of the data.

The first quartile (Q1) is the 25th percentile.

The second quartile (Q2) is the 50th percentile, that is, the median.

The third quartile (Q3) is the 75th percentile.

For small data sets, the division may be into four parts of only approximately equal size.

Example 3.35 - Finding the Example 3.35 - Finding the quartiles for a small data set: quartiles for a small data set: the dance audition scoresthe dance audition scoresIn Example 3.24 (page 126) we examined the

dance scores of 12 students auditioning

for admission into a prestigious graduate

school of the arts. Recall that we found the

75th percentile of the dance audition scores to

be 83. By definition, the 75th percentile

is the third quartile Q3. Therefore, this score of 83

is also the third quartile (Q3) of the audition

scores. Now we will find the first quartile and the

median (second quartile).

Example 3.35 continuedExample 3.35 continued

FIGURE 3.34 The quartiles for the dance audition data.

Interquartile RangeInterquartile Range

Interquartile range (IQR) is a robust measure of variability.

IQR = Q3 - Q1

The interquartile range is interpreted to be the spread of the middle 50% of the data.

Example 3.37 - Finding the Example 3.37 - Finding the interquartile range for the dance interquartile range for the dance audition scoresaudition scores

In Example 3.35, we found that, for the dance audition score data, Q1 = 59 and Q3 = 83.Find the IQR for the dance score data and explain what it means.

Example 3.37 continuedExample 3.37 continued

Solution

We would say that the middle 50%, or middle half, of the dance audition scores ranged over 24 points (see Figure 3.38).

FIGURE 3.38 The IQR for the dance audition data.

Example 3.37 continuedExample 3.37 continued

What would happen if we introduced an outlier into this data set?

Change the lowest score from 30 to 3?

IQR completely unaffected even if we changed the 44 to a 4.

f we changed the 56, then the IQR would be affected, since Q1 would then change.

The Five-Number SummaryThe Five-Number Summary

Consists of the following set of statistics, which together constitute a robust summarization of a data set:

1. Smallest value in the data set (minimum)

2. First quartile, Q1

3. Median, Q2

4. Third quartile, Q3

5. Largest value in the data set (maximum)

Example 3.38 - Find the five-Example 3.38 - Find the five-number summary for the dance number summary for the dance

audition data.audition data.Solution:

Figure 3.39 shows the five-number summary

FIGURE 3.39 The quartiles for the dance audition data.

Example 3.38 continuedExample 3.38 continued

1. Minimum = 30

2. First quartile, Q1 = 59

3. Median Q2 = 71.5

4. Third quartile, Q3 = 83

5. Maximum = 94

Example 3.38 continuedExample 3.38 continued

The five-number summary is often reported as Min = 30, Q1 = 59, Med = 71.5, Q3 = 83, Max = 94.

Which parts of the five-number summary are less robust than others?

Since the minimum and maximum are the most extreme values, these are clearly very sensitive to outliers.

However, Q1, the median, and Q3 are very resistant to the influence of outliers.

The BoxplotThe Boxplot

Convenient graphical display of the five-number summary

Allows the data analyst to evaluate the symmetry or skewness of a data set.

Example 3.41 - Constructing a Example 3.41 - Constructing a Boxplot by handBoxplot by hand

Construct a boxplot for the dance score data.

Use the steps for constructing a boxplot on page 148.

Example 3.41 continuedExample 3.41 continuedSolution From Example 3.38, the five-number summary

for the dance score data is

Min = 30, Q1 = 59, Med = 71.5, Q3 = 83, Max = 94.

From Example 3.37, the interquartile

range for the dance score data is

IQR = Q3 - Q1 = 83 - 59 = 24.

Step 1: Determine the lower and upper fences:

a. Lower fence Q1 - 1.5(IQR)

= 59 - 1.5(24) = 59 - 36 = 23

b. Upper fence Q3 - 1.5(IQR)

= 83 + 1.5(24) = 83 + 36 = 119

Example 3.41 continuedExample 3.41 continued

Step 2: Draw a horizontal number line that encompasses the range of your data,including the fences. Above the number line, draw vertical lines

at Q1 = 59, median = 71.5, and Q3 = 83. Connect the lines for Q1 and Q3 to each

other so as to form a box, as shown in Figure 3.41a below.

Example 3.41 continuedExample 3.41 continued

Step 3:Temporarily indicate the fences (lower fence 23 and upper fence 119) as brackets above the number line. (See Figure 3.41b below.)

Example 3.41 continuedExample 3.41 continuedStep 4:Draw a horizontal line from Q1 = 59 to thesmallest data value greater than the lower fence. The lowest data value is Min = 30.

This is greater than the lower fence = 23.

So draw the line from 59 to 23.

Draw a horizontal line from Q3 = 83 to the largest data value smaller than the upper fence.

Example 3.41 continuedExample 3.41 continued

The largest data value is Max = 94, which is smaller than the upper fence.

So draw the line from 83 to 94. (See Figure 3.41c below.)

Example 3.41 continuedExample 3.41 continuedStep 5:There are no data values lower than the lower fence or greater than the upper fence.

No outliers in this data set.

Remove the temporary brackets. See Figure 3.41d below.

Robust Detection of OutliersRobust Detection of Outliers

Use a five-number summary or a boxplot to detect outliers, as follows:

A data value is an outlier if

a) It is located 1.5(IQR) or more below Q1, or

b) It is located 1.5(IQR) or more above Q3.

Example 3.45 - Robust Example 3.45 - Robust detection of outliers for the detection of outliers for the dance audition datadance audition data

Determine if there are any outliers in thedance score data.

Example 3.45 continuedExample 3.45 continued

Solution

Recall for the dance score data set that

IQR = 24, Q1 = 59, and Q3 = 83.

So we have 1.5(IQR) = 1.5(24) = 36.

Q1 – 1.5(IQR) and Q3 + 1.5(IQR):

The first step is to find the two quantities

Q1 – 1.5(IQR) = Q1 – 36 = 59 – 36 = 23

Q3 + 1.5(IQR) = Q3 + 36 = 83 + 36 = 119

Example 3.45 continuedExample 3.45 continued

No data values less then or equal to 23 or greater than or equal to 119.

No outliers are identified by the robust method.

SummarySummary

Section 3.6 presents robust measures and methods, which are not sensitive to the presence of outliers.

Quartiles divide the data set into approximately equal quarters.

The interquartile range is a measure of variability found by taking the difference between the third and first quartiles.

SummarySummary

The five-number summary is a robust alternative to the usual mean-and-standard- deviation method of summarizing a data set.

It consists of simply reporting the minimum, first quartile, median, third quartile, and maximum of the data set.

SummarySummaryA boxplot is a graphical representation of the

five-number summary and is useful for investigating skewness and the presence of outliers.

The robust method of detecting outliers is to consider a data value an outlier if it is located 1.5(IQR) or more below Q1, or it is located 1.5(IQR) or more above Q3.