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Chapter 3

Measures of Position

Seven edition - Elementary statistic

Bluman

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Measures of Position

1. Z-score

2. Percentile

3. Quartile

4. Outlier

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Introduction

In addition to measures of central tendency and

measures of variation, there are measures of

position or location. These measures include

standard scores, percentiles, deciles, and quartiles.

They are used to locate the relative position of a data

value in the data set.

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Introduction(cont’d)

Examples

If a value is located at the 80th percentile, it means that

80% of the values fall below it in the distribution and

20% of the values fall above it.

The median is the value that corresponds to the 50th

percentile, since one-half of the values fall below it and

one-half of the values fall above it.

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Z-score or standard score

•

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Note that if the z score is positive, the score is above the

mean. If the z score is 0, the score is the same as the

mean. And if the z score is negative, the score is below

the mean.

When all data for a variable are transformed into z

scores, the resulting distribution will have a mean of 0

and a standard deviation of 1.

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Percentiles

7- 10The class

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Finding a Data Value Corresponding to a

Given Percentile

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Finding a Data Value Corresponding to

a Given Percentile

Step 3

If c is not a whole number, round it up to the next whole

number; in this case, c = 3.Start at the lowest value and

count over to the third value, which is 5. Hence, the value

5 corresponds to the 25th percentile.

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2, 3, 5, 6, 8, 10, 12, 15, 18, 20

6th value 7th value

The value halfway between 10 and 12 is 11. Find it by

adding the two values and dividing by 2.

Hence, 11 corresponds to the 60th percentile. Anyone

scoring 11 would have done better than 60% of the class.

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Quartiles and Deciles

Quartiles: divide the distribution into four groups,

separated by Q1, Q2, Q3.

Note that :

Q1 is the same as the 25th percentile;

Q2 is the same as the 50th percentile, or the median;

Q3 corresponds to the 75th percentile, as shown

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Calculate the percentile

Finding Data Values Corresponding to Q1, Q2, and Q3

Step 1 Arrange the data in order from lowest to highest.

Step 2 Find the median of the data values. This is the value

for Q2.

Step 3 Find the median of the data values that fall below

Q2. This is the value for Q1.

Step 4 Find the median of the data values that fall above

Q2. This is the value for Q3.

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Quartiles is found in this procedure table

Procedure Table

Finding Data Values Corresponding to Q1, Q2, and Q3

Step1 Arrange the data in order from lowest to highest

Step2 Find the median of the data values. This is the value for Q2.

Step3 Find the median of the data values that fall below Q2. This is the value for Q1.

Step4 Find the median of the data values that fall above Q2. This is the value for Q3.

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Summary of Position Measures

Measure Definition

Standard score Number of standard deviations that a data value is above or below the

mean

Percentile Position in hundredths that a data value holds in the distribution

Decile Position in tenths that a data value holds in the distribution

Quartile Position in fourths that a data value holds in the distribution

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Outliers

An outlier is an extremely high or an extremely low data

value when compared with the rest of the data values.

An outlier can strongly affect the mean and standard

deviation of a variable.

This value would then make the mean and standard

deviation of the variable much larger than they really

were.

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Procedure Table

Procedure for Identifying Outliers

Step1 Arrange the data in order and find Q1 and Q3.

Step2 Find the interquartile range: IQR =Q3 - Q1.

Step3 Multiply the IQR by 1.5.

Step4 Subtract the value obtained in step 3 from Q1 and add the value to Q3

step5 Check the data set for any data value that is smaller than Q1 - 1.5(IQR) or

larger than Q3 +1.5(IQR).

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Example: Check the following data set for outliers.

5, 6, 12, 13, 15, 18, 22, 50

Solution

The data value 50 is extremely suspect. These are the steps

in checking for an outlier.

Step 1 Find Q1 and Q3.

Q1 is 9 and Q3 is 20.

Step 2 Find the interquartile range (IQR), which is

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Q3 -Q1. , IQR = Q3 - Q1 =20 -9 = 11

Step 3 Multiply this value by 1.5.

1.5(11) =16.5

Step 4 Subtract the value obtained in step 3 from Q1, and

add the value obtained in step 3 to Q3.

9-16.5= 7.5 and 20 + 16.5 = 36.5

Step 5 Check the data set for any data values that fall

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outside the interval from 7.5 to 36.5.

The value 50 is outside this interval; hence, it can be

considered an outlier.

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There are several reasons why outliers may

occur

The data value may have resulted from a measurement

or observational error. Perhaps the researcher measured

the variable incorrectly.

The resulted from a recording error. That is, it may have

been written or typed incorrectly.

The data value may have been obtained from a subject

that is not in the defined population.

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Boxplot

A boxplot is a graph of a data set obtained by drawing a

horizontal line from the minimum data value to Q1,

drawing a horizontal line from Q3 to the maximum data

value, and drawing a box whose vertical sides pass

through Q1 and Q3 with a vertical line inside the box

passing through the median or Q2.

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Procedure for constructing a

boxplot

1. Find the five-number summary for the data values,

that is, the maximum and minimum data values, Q1 and

Q3, and the median.

2. Draw a horizontal axis with a scale such that it

includes the maximum and minimum data values.

3. Draw a box whose vertical sides go through Q1 and

Q3, and draw a vertical line though the median.

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4. Draw a line from the minimum data value to the left

side of the box and a line from the maximum data value to

the right side of the box.

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The number of meteorites found in 10 states of the United

States is 89, 47, 164, 296, 30, 215, 138, 78, 48, 39.

Construct a boxplot for the data.

Step 1 Arrange the data in order.

Step 2 Find the median Q2=83.5.

Step 3 Find Q1=47.

Step4 Find Q3=164.

Step 5 Draw a scale for the data on the x axis.

47 83.5 164

30 296

The distribution is somewhat positively skewed

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If the median falls to the left of the center of the box, the

distribution is positively skewed.

If the right line is larger than the left line, the distribution

is positively skewed.

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Information obtained from a boxplot

If the median falls to the right of the center, the

distribution is negatively skewed. If the left line is larger

than the right line, the distribution is negatively skewed.

If the median is near the center of the box, the

distribution is approximately symmetric.

If the lines are about the same length, the distribution is