SECTION 3.3: MEASURES OF POSITION

OBJECTIVES

1. Compute and interpret z-scores2. Compute the quartiles of a data set3. Compute the percentiles of a data set4. Compute the five-number summary for a data set5. Understand the effects of outliers6. Construct boxplots to visualize the five-number summary and outliers

OBJECTIVE 1COMPUTE AND INTERPRET z-SCORES

Z-SCORE

Who is taller, a man 73 inches tall or a woman 68 inches tall? The obvious answer is that the man is taller. However, men are taller than women on the average. Suppose the question is asked this way: Who is taller relative to their gender, a man 73 inches tall or a woman 68 inches tall? One way to answer this question is with a z-score.

The z-score of an individual data value tells how many _____________________________ that

value is from its population mean. For example, a value one standard deviation above the

mean has a z-score of ____________ and a value two standard deviations below the mean has

a z-score of ____________.

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Let x be a value from a population with mean μ and standard deviation σ .

The z-score for xis z = ❑ .

SECTION 3.3: MEASURES OF POSITION

EXAMPLE: A National Center for Health Statistics study states that the mean height for adult men in the U.S. is μ = 69.4 inches, with a standard deviation of σ = 3.1 inches. The mean height for adult women is μ = 63.8 inches, with a standard deviation of σ = 2.8 inches. Who is taller relative to their gender, a man 73 inches tall, or a woman 68 inches tall?

SOLUTION:

Z-SCORES & THE EMPIRICAL RULE

Since the z-score is the number of standard deviations from the mean, we can easily interpret the z-score for bell-shaped populations using The Empirical Rule.When a population has a histogram that is approximately bell-shaped, then• Approximately 68% of the data will have z-scores between –1 and 1.• Approximately 95% of the data will have z-scores between –2 and 2.• All, or almost all of the data will have z-scores between –3 and 3.

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SECTION 3.3: MEASURES OF POSITION

OBJECTIVE 2COMPUTE THE QUARTILES OF A DATA SET

QUARTILES

In a previous section, we learned how to compute the mean and median of a data set as measures of the center. Sometimes, it is useful to compute measures of position other than the center to get a more detailed description of the distribution. Quartiles divide a data set into four approximately equal pieces.

QUARTILES:

Every data set has three quartiles:

The __________________________, denoted Q1 separates the lowest __________ of the data

from the highest __________.

The __________________________, denoted Q2 separates the lowest __________ of the data

from the highest __________. Q2 is the same as the median.

The __________________________, denoted Q3 separates the lowest __________ of the data

from the highest __________.

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SECTION 3.3: MEASURES OF POSITION

COMPUTING QUARTILES

There are several methods for computing quartiles, all of which give similar results. The following procedure is one fairly straightforward method:

Step 1: Arrange the data in increasing order.Step 2: Let n be the number of values in the data set. To compute the second quartile,

simply compute the median. For the first or third quartiles, proceed as follows:For the first quartile, compute L=0.25nFor the third quartile, compute L=0.75n

Step 3: If L is a whole number, the quartile is the average of the number in position L and the number in position L+1.If L is not a whole number, round it up to the next higher whole number. The quartile is the number in the position corresponding to the rounded-up value.

EXAMPLE: The following table presents the annual rainfall, in inches, in Los Angeles during the month of February from 1969 to 2013. Compute the quartiles for the data.

0.00 0.08 0.13 0.14 0.1 0.17 0.20 0.29 0.56 0.67 0.70 0.92 1.22 1.30 1.48

1.64 1.72 1.90 2.37 2.5 2.84 3.06 3.12 3.21 3.29 3.54 3.57 3.71 4.13 4.27

4.37 4.64 4.89 4.94 5.5 6.10 6.61 7.89 7.96 8.03 8.87 8.91 11.0 12.75 13.68

SOLUTION:

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SECTION 3.3: MEASURES OF POSITION

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SECTION 3.3: MEASURES OF POSITION

QUARTILES ON THE TI-84 PLUS

The 1-Var Stats command in the TI-84 PLUS Calculator displays a list of the most common parameters and statistics for a given data set. This command is accessed by pressing STAT and then highlighting the CALC menu.

EXAMPLE: QUARTILES ON THE TI-84 PLUS

Step 1: Enter the data in L1.

Step 2: Press STAT and highlight the CALC menu.

Step 3: Select 1-Var Stats and press ENTER. Enter L1 in the List field and run the command.

Note: If your calculator does not support Stat Wizards, enter L1 next to the 1-Var Stats command on the home screen and press enter to run the command

The quartile values produced by the TI-84 PLUS may differ from results obtained by hand because it uses a slightly different procedure than the one described in the text.

VISUALIZING THE QUARTILES

Following is a dotplot of the Los Angeles rainfall data with the quartiles indicated. The quartiles divide the data set into four parts, with approximately 25% of the data in each part.

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SECTION 3.3: MEASURES OF POSITION

OBJECTIVE 3COMPUTE THE PERCENTILES OF A DATA SET

Quartiles describe the shape of a distribution by dividing it

into fourths. Sometimes it is useful to divide a data set into

a greater number of pieces to get a more detailed

description of the distribution.

________________________ divide a data set into

hundredths.

PERCENTILES:

For a number p between 1 and 99, the pth percentile separates the lowest p% of the data from the highest (1– p)%.

COMPUTING PERCENTILES

The following procedure computes the pth percentile of a data set:Step 1: Arrange the data in increasing order.Step 2: Let n be the number of values in the data set. For the pth percentile, compute

L=( p100 )n.

Step 3: If L is a whole number, the pth percentile is the average of the number in position L and the number in position L+1.If L is not a whole number, round it up to the next higher whole number. The pth

percentile is the number in the position corresponding to the rounded-up value.

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SECTION 3.3: MEASURES OF POSITION

EXAMPLE: The following table presents the annual rainfall, in inches, in Los Angeles during the month of February from 1969 to 2013. Compute the 60th percentile for the data.

0.00 0.08 0.13 0.14 0.1 0.17 0.20 0.29 0.56 0.67 0.70 0.92 1.22 1.30 1.48

1.64 1.72 1.90 2.37 2.5 2.84 3.06 3.12 3.21 3.29 3.54 3.57 3.71 4.13 4.27

4.37 4.64 4.89 4.94 5.5 6.10 6.61 7.89 7.96 8.03 8.87 8.91 11.0 12.75 13.68

SOLUTION:

COMPUTING A PERCENTILE FROM A GIVEN DATA VALUE

Sometimes we are given a value from a data set and wish to compute the percentile corresponding to that value. Following is the procedure for doing this:

Step 1: Arrange the data in increasing order.Step 2: Let x be the data value whose percentile is to be computed. Use the following

formula to compute the percentile:

Percentile = 100 ∙ ( Number of values less than x )+0.5Number of values∈the dataset

Round the result to the nearest whole number. This is the percentile corresponding to the value x.

EXAMPLE: The following table presents the annual rainfall in Los Angeles during February from 1969 to 2013. In 1989, the rainfall was 1.90. What percentile does this correspond to?

0.00 0.08 0.13 0.14 0.1 0.17 0.20 0.29 0.56 0.67 0.70 0.92 1.22 1.30 1.48

1.64 1.72 1.90 2.37 2.5 2.84 3.06 3.12 3.21 3.29 3.54 3.57 3.71 4.13 4.27

4.37 4.64 4.89 4.94 5.5 6.10 6.61 7.89 7.96 8.03 8.87 8.91 11.0 12.75 13.68

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SECTION 3.3: MEASURES OF POSITION

SOLUTION:

OBJECTIVE 4COMPUTE THE FIVE-NUMBER SUMMARY FOR A DATA SET

The ___________________________________________ of a data set consists of the median, the first quartile, the third quartile, the smallest value, and the largest value. These values are generally arranged in order.

EXAMPLE: Recall the Los Angeles annual rainfall data. Compute the five-number summary.0.00 0.08 0.13 0.14 0.1 0.17 0.20 0.29 0.56 0.67 0.70 0.92 1.22 1.30 1.48

1.64 1.72 1.90 2.37 2.5 2.84 3.06 3.12 3.21 3.29 3.54 3.57 3.71 4.13 4.27

4.37 4.64 4.89 4.94 5.5 6.10 6.61 7.89 7.96 8.03 8.87 8.91 11.0 12.75 13.68

SOLUTION:

When using the TI-84 PLUS Calculator, the five-number summary is given by the 1-Var Stats command.

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SECTION 3.3: MEASURES OF POSITION

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SECTION 3.3: MEASURES OF POSITION

OBJECTIVE 5UNDERSTAND THE EFFECTS OF OUTLIERS

An _______________ is a value that is considerably larger or considerably smaller than most of the values in a data set. Some outliers result from errors; for example a misplaced decimal point may cause a number to be much larger or smaller than the other values in a data set. Some outliers are correct values, and simply reflect the fact that the population contains some extreme values.

EXAMPLE: The temperature in a downtown location is measured for eight consecutive days during the summer. The readings, in Fahrenheit, are

81.2 85.6 89.3 91.0 83.2 8.45 79.5 87.8Which reading is an outlier? Is the outlier an error or is it possible that it is correct?

SOLUTION:

INTERQUARTILE RANGE

One method for detecting outliers involves a measure called the Interquartile Range.

INTERQUARTILE RANGE

The interquartile range is found by subtracting the __________ quartile from the __________ quartile.

IQR = __________________________

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SECTION 3.3: MEASURES OF POSITION

IQR METHOD FOR DETECTING OUTLIERS

The most frequent method used to detect outliers in a data set is the IQR Method. The procedure for the IQR Method is:Step 1: Find the first quartile Q1, and the third quartile Q3.Step 2: Compute the interquartile range: IQR = Q3−Q1.Step 3: Compute the outlier boundaries. These boundaries are the cutoff points for

determining outliers:Lower Outlier Boundary = Q1 – 1.5(IQR)Upper Outlier Boundary = Q3 + 1.5(IQR)

Step 4: Any data value that is less than the lower outlier boundary or greater than the upper outlier boundary is considered to be an outlier.

EXAMPLE: The following table presents the number of students absent in a middle school in northwestern Montana for each school day in January. Identify any outliers.65 67 71 57 51 49 44 41 59 49 42 5645 77 44 42 45 46 100 59 53 51

SOLUTION:

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SECTION 3.3: MEASURES OF POSITION

OBJECTIVE 6CONSTRUCT BOXPLOTS TO VISUALIZE THE FIVE-NUMBER SUMMARY AND OUTLIERS

A _____________________ is a graph that presents the five-number summary along with some

additional information about a data set. There are several different kinds of boxplots. The one

we describe here is sometimes called a __________________________________.

EXAMPLE: The following table presents the number of students absent in a middle school in northwestern Montana for each school day in January. Identify any outliers.65 67 71 57 51 49 44 41 59 49 42 5645 77 44 42 45 46 100 59 53 51

SOLUTION:

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SECTION 3.3: MEASURES OF POSITION

BOXPLOTS ON THE TI-84 PLUS

The following steps will create a boxplot for the student absences data on the TI-84 PLUS.

Step 1: Enter the data in L1.

Step 2: Press 2nd,Y=, then 1 to access the Plot1 menu. Select On and the boxplot type.

Step 3: Press Zoom, 9 to view the plot.

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SECTION 3.3: MEASURES OF POSITION

DETERMINING THE SHAPE OF A DATA SET FROM A BOXPLOT

Boxplots can be used to determine skewness in a data set.

If the median is closer to the first quartile than to the third quartile, or the upper whisker is longer than the lower whisker, the data are skewed to the right.

If the median is closer to the third quartile than to the first quartile, or the lower whisker is longer than the upper whisker, the data are skewed to the left.

If the median is approximately halfway between the first and third quartiles, and the two whiskers are approximately equal in length, the data are approximately symmetric

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SECTION 3.3: MEASURES OF POSITION

YOU SHOULD KNOW …

How to compute and interpret z-scores

How to compute the quartiles of a data set

How to compute a percentile of a data set

How to compute the percentile corresponding to a given data value

How to find the five-number summary for a data set

How to determine outliers using the IQR method

How to construct a boxplot and use it to determine skewness

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