Section 3.4 Measures of Relative Standing

1. z-scores

2. percentiles

3. Quartiles (Q1, Q3)

4. interquartile range (IQR)

5. Identifying outliers

3-1

z Score (or standardized value)

= the number of standard deviations that a

given value x is above or below the mean

Z score:

Sample Population

x - µz =

Round z to 2 decimal places

z = x - xs

Z score:

Interpreting Z Scores:

Whenever a value is less than the mean, its corresponding z score is negative

Ordinary values: z score between –2 and 2 Unusual Values: z score < -2 or z score > 2

EXAMPLE

The mean height of males 20 years or older is 69.1 inches with a standard deviation of 2.8 inches. The mean height of females 20 years or older is 63.7 inches with a standard deviation of 2.7 inches. Data based on information obtained from National Health and Examination Survey. Who is relatively taller?

Kevin Garnett whose height is 83 inches

or

Candace Parker whose height is 76 inches

3-5

83 69.1

2.84.96

kgz

76 63.7

2.74.56

cpz

Kevin Garnett’s height is 4.96 standard deviations above the mean. Candace Parker’s height is 4.56 standard deviations above the mean. Kevin Garnett is relatively taller.

3-6

Percentiles:

Just as there are three quartiles separating data into four parts, there are 99 percentiles denoted P1, P2, . . . P99, which partition the data into 100 groups.

The kth percentile, denoted, Pk, of a set of data is a value such that k percent of the observations are less than or equal to the value.

3-8

Finding the Percentile of a Given Score

Percentile of value x = • 100number of values less than x

total number of values

EXAMPLE Interpret a Percentile

The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. The University of Pittsburgh Graduate School of Public Health requires a GRE score no less than the 70th percentile for admission into their Human Genetics MPH or MS program.

(Source: http://www.publichealth.pitt.edu/interior.php?pageID=101.)

Interpret this admissions requirement.

In general, the 70th percentile is the score such that 70% of the individuals who took the exam scored worse, and 30% of the individuals scores better. In order to be admitted to this program, an applicant must score as high or higher than 70% of the people who take the GRE. Put another way, the individual’s score must be in the top 30%.

3-10

Another Example:

• There are 125 people in a class, and 95 of them are less tall than you. What is your percentile in height among the whole class?

Answer: 95 = (k/100) * 125 Solve for k: k = 76 Your ‘position’ in height is at the 76th percentile.

Or (more straightforward):

Compute the fraction:

Convert 0.76 to percent: 76% -> ‘position’ in height is at the 76th percentile

76.0125

95

Quartiles:

Q1 (First Quartile)

separates the bottom 25% of sorted values from the top 75%.

Q2 (Second Quartile)

same as the median; separates the bottom 50% of sorted values from the top 50%.

Q1 (Third Quartile)

separates the bottom 75% of sorted values from the top 25%.

Q1, Q2, Q3:

divide ranked scores into four equal parts:

Quartiles:

25% 25% 25% 25%

Q3Q2Q1(min) (max)

(median)

3-14

A group of Brigham Young University—Idaho students (Matthew Herring, Nathan Spencer, Mark Walker, and Mark Steiner) collected data on the speed of vehicles traveling through a construction zone on a state highway, where the posted speed was 25 mph. The recorded speed of 14 randomly selected vehicles is given below:

20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40

Find and interpret the quartiles for speed in the construction zone.

EXAMPLE Finding and Interpreting Quartiles

Step 1: The data is already in ascending order.

Step 2: There are n = 14 observations, so the median, or second quartile, Q2, is the mean of the 7th and 8th observations. Therefore, M = 32.5.

Step 3: The median of the bottom half of the data is the first quartile, Q1.

20, 24, 27, 28, 29, 30, 32

The median of these seven observations is 28. Therefore, Q1 = 28. The median of the top half of the data is the third quartile, Q3. Therefore, Q3 = 38.

3-15

Interpretation:

• 25% of the speeds are less than or equal to the first quartile, 28 miles per hour, and 75% of the speeds are greater than 28 miles per hour.

• 50% of the speeds are less than or equal to the second quartile, 32.5 miles per hour, and 50% of the speeds are greater than 32.5 miles per hour.

• 75% of the speeds are less than or equal to the third quartile, 38 miles per hour, and 25% of the speeds are greater than 38 miles per hour.

3-16

Interquartile Range (or IQR):

IQR = Q3 - Q1

Interquartile Range (IQR):

EXAMPLE Determining and Interpreting the Interquartile Range

Determine and interpret the interquartile range of the speed data.

Q1 = 28 Q3 = 38

3 1IQR

38 28

10

Q Q

The range of the middle 50% of the speed of cars traveling through the construction zone is 10 miles per hour.

3-18

Suppose a 15th car travels through the construction zone at 100 miles per hour. How does this value impact the mean, median, standard deviation, and interquartile range?

Without 15th car With 15th car

Mean 32.1 mph 36.7 mph

Median 32.5 mph 33 mph

Standard deviation 6.2 mph 18.5 mph

IQR 10 mph 11 mph

-19

3-20

Identifying Outliers

The length between the fence and quartile is called: Whisker

EXAMPLE

Check the speed data for outliers.

Step 1: The first and third quartiles are Q1 = 28 mph and Q3 = 38 mph.

Step 2: The interquartile range is 10 mph.

Step 3: The fences are

Lower fence= Q1 – 1.5(IQR) Upper fence= Q3 + 1.5(IQR)

= 28 – 1.5(10) = 38 + 1.5(10)

= 13 mph = 53 mph

Step 4: There are no values less than 13 mph or greater than 53 mph. Therefore, there are no outliers.

Section 3.5 The Five-Number Summary and Boxplots

1. The five-number summary

2. Boxplots

3-22

For a set of data, the 5-number summary consists of minimum value; the first quartile Q1;

the median (or second quartile Q2);

the third quartile, Q3; and

the maximum value.

Boxplot (or box-and-whisker-diagram):

= graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q1; the median; and the third quartile, Q3.

The 5-number summary and Boxplots

Boxplot:

• Boxplots are a good way to identify outliers.Boxplots are a good way to identify outliers.• IQR IQR stands for:stands for: I Internter Q Quartileuartile R Range. ange. • IQR Boxplots are also called IQR Boxplots are also called box-and-whiskersbox-and-whiskers plots. plots.

• Definitions: Definitions: A box plotA box plot

First quartile

(25% position in data)

Also called: ‘hinge’

Third quartile

(75% position in data)

Also called: ‘hinge’Middle half of the data

Left whisker

MedianRight whisker

Boxplot: extended

• The length of the whiskers is determined by the following The length of the whiskers is determined by the following calculations:calculations:

1.1. IQR = Third Quartile – First QuartileIQR = Third Quartile – First Quartile

2.2. Left whisker = First Quartile – 1.5 Left whisker = First Quartile – 1.5 timestimes (IQR) (IQR)

3.3. Right whisker = Third Quartile + 1.5 Right whisker = Third Quartile + 1.5 timestimes (IQR) (IQR)

• Outliers are represented by an asterisk (*)Outliers are represented by an asterisk (*)

*

Outlier

EXAMPLE Five-Number Summary

Every six months, the United States Federal Reserve Board conducts a survey of credit card plans in the U.S. The following data are the interest rates charged by 10 credit card issuers randomly selected for the July 2005 survey. Determine the five-number summary of the data.

Institution Rate

Pulaski Bank and Trust Company 6.5%

Rainier Pacific Savings Bank 12.0%

Wells Fargo Bank NA 14.4%

Firstbank of Colorado 14.4%

Lafayette Ambassador Bank 14.3%

Infibank 13.0%

United Bank, Inc. 13.3%

First National Bank of The Mid-Cities 13.9%

Bank of Louisiana 9.9%

Bar Harbor Bank and Trust Company 14.5%

Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm

First, we write the data is ascending order:

6.5%, 9.9%, 12.0%, 13.0%, 13.3%, 13.9%, 14.3%, 14.4%, 14.4%, 14.5%

The smallest number is 6.5%. The largest number is 14.5%. The first quartile is 12.0%. The second quartile is 13.6%. The third quartile is 14.4%.

Five-number Summary:

6.5% 12.0% 13.6% 14.4% 14.5%

Step 1: The interquartile range (IQR) is 14.4% - 12% = 2.4%. The lower and upper fences are:

Lower Fence = Q1 – 1.5(IQR) Upper Fence = Q3 + 1.5(IQR)

= 12 – 1.5(2.4) = 14.4 + 1.5(2.4)

= 8.4% = 18.0%

Step 2:

[ ]*

EXAMPLE Boxplot

In the Example of The interest rate: The boxplot indicates that the distribution is skewed left.

Using boxplots and quartiles to describe the shape of a distribution

For your Practice…

Try these problems!(answers are included)

Exercise 1:Here are 37 home prices in different neighborhoods of Chicago. Are any of the home prices

suspicious (outliers)? $123,900 $130,900 $133,900 $138,900 $139,900 $146,900 $156,900 $156,900 $158,900 $159,400 $160,900 $163,900 $167,900 $167,900 $176,900 $182,900 $184,900 $186,900 $199,900 $199,900 $200,900 $204,900 $219,900 $219,900 $254,900 $256,300 $292,000 $311,750 $369,900 $385,500 $410,300 $430,500 $431,700 $436,500 $487,500 $496,500 $556,800

Answer: Build a Boxplot

• Find the Median:

• Find the first quartile:

• Find the third quartile:

• Calculate IQR:

• Calculate the whiskers: left whisker =

right whisker =

• Draw the boxplot, include all 5 numbers (median, quartiles, whiskers) on it.

• Identify outliers if any.

$14,655 $14,799

$15,605 $16,395 $16,798 $17,990 $19,300 $20,000

$21,995 $22,195 $22,708 $23,240 $23,405 $23,920 $25,176

$25,999$26,185 $26,268 $27,815 $27,910 $28,680 $28,950 $29,099

$29,249$30,585 $30,645 $31,985 $32,250 $32,950 $33,595 $33,790

$34,590$35,550 $36,300 $38,175 $41,188 $42,660 $54,950 $56,000

$63,500

20

ob

serv

atio

ns

20

ob

serv

atio

ns

Median = ($27,910+$28,680)/2 = $28,295

First quartile: Q1 = ($22,195+$22,708)/2 = $22,451.50

Third quartile: Q3 = ($33,595+$33,790)/2 = $33,692.50

• Order the data

• Find the Median and the first and third quartiles

10 o

bser

vatio

ns10

obs

erva

tions

10 o

bser

vatio

ns10

obs

erva

tions

• Now we can calculate the ‘whiskers’ and build the boxplot

Answer

• Calculate the whiskers:

1. IQR = Third Quartile – First Quartile = $33.692.50 – $22,451.50 = $11,241.00

2. 1.5 times (IQR) = 1.5($11,241.00) = $16,861.50

3. Left Whisker = First Quartile – 1.5 times (IQR) =  $22,451.50 – $16,861.50 = $5,590.00

4. Right Whisker = Third Quartile + 1.5 times (IQR) =    $33,692.50 + $16,861.50 = $50,554.00

• Build the Boxplot:

OUTLIERS: The three asterisks represent $54,950 $56,000 and $63,500 which are unusually high compared to the other prices in the data set. These prices should be investigated before further analysis of the data is performed.

Exercise 2:Data for 37 home prices in different neighborhoods of Chicago. $123,900 $130,900 $133,900 $138,900 $139,900 $146,900 $156,900 $156,900 $158,900 $159,400 $160,900 $163,900 $167,900 $167,900 $176,900 $182,900 $184,900 $186,900 $199,900 $199,900 $200,900 $204,900 $219,900 $219,900 $254,900 $256,300 $292,000 $311,750 $369,900 $385,500 $410,300 $430,500 $431,700 $436,500 $487,500 $496,500 $556,800

Answers

•Median: (n+1)/2=19; the 19th observation is the median = $199,900

Important: The Median is not used in calculating the quartiles (next)!

• First quartile: it’s the middle of the 18 observations below median = ($158,900+$159,400)/2= $159.150

• Third quartile: it’s the middle of the 19 observations above median= ($311,750+ $369,900)/2 = $340,825

• IQR: IQR = $340,825 – $159,150 = $181,675

• Whiskers: left whisker = $159,150- 1.5 x $181,675 = negative so choose $0

right whisker = $340,825+1.5 x $181,675 = $613,337.50

• Boxplot:

• Identify outliers if any: none in this case.

Exercise 3These are prices for new 1100cc to 1200 cc motorcycle:

$14,500 $15,600 $16,700 $ 8,990 $15,000 $16,990 $18,200 $19,200$16,190 $11,999 $11,999 $ 7,399 $10,449 $12,299 $ 8,995 $ 7,895 $ 8,595 $ 7,399 $ 7,999 $ 8,199 $ 9,999 $ 9,699 $10,599 $8,399$ 7,899 $10,899 $ 6,999 $14,399 $11,770

Answers

Answers: (Using calculator or Excel)

•Mean=$11,560.65, Median=$10,599

•First quartile=$8,299

•Third quartile=$14,750

IQR = $14,750 - $8,299 = $6451

Left whisker = $8,299- 1.5($6451) = $1,377.5

Right whisker = $14,750 +1.5($6451) = $24,426.5

Box plot:

No outliers.

$1,377.5 $24,426.5

$8,299 $14,750

$10,599