Chapter 2Describing Data with

Numerical Measurements

General Objectives:

Graphs are extremely useful for the visual description of a data set. However, they are not always the best tool when you want to make inferences about a population from the information contained in a sample. For this purpose, it is better to use numerical measures to construct a mental picture of the data.

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Specific Topics1. Measures of center: mean, median, and mode

2. Measures of variability: range, variance, and standard deviation

3. Tchebysheff’s Theorem and the Empirical Rule

4. Measures of relative standing: z-scores, percentiles, quartiles, and the interquartile range

5. Box plots

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2.1 and 2.2 Describing a Set of Data with Numerical Measures

and Measures of Center

Definition: Numerical descriptive measures associated with a population of measurements are called parameters.

Those computed from sample measurements are called statistics.

Definition: The arithmetic mean or average of a set is equal to the sum of the measurements divided by n.

Notation:

Sample mean:

Population mean:

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_x

Example 2.1

Use a dotplot to display the n 5 measurements 2, 9, 11, 5, 6.Find the sample mean of these observations, and compare its value with what you might consider the “center” of these observations on the dotplot.

Solution

The dotplot in Figure 2.2 seems to be centered between 6 and 8. To find the sample mean, we can calculate

The statistic 6.6 is the balancing point or fulcrum shown on the dotplot. It does seem to mark the center of the data.

Figure 2.2 Character Dotplot

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6.65

651192 nx

x i

x

Definition: The median m of a set of n measurements is the value of x that falls in the middle position when the measurements are ordered from smallest to largest.

The median is less sensitive to extreme values or outliers than the mean.

The value .5(n + 1) indicates the position of the median in the ordered data set.

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©1998 Brooks/Cole Publishing/ITP

©1998 Brooks/Cole Publishing/ITP

Figure 2.3(a) Relative frequency distribution showing the effect of extreme values on the mean and median

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Figure 2.3(b)

Definition: The mode is the category that occurs most frequently, or the most frequently occurring value of x.

When measurements on a continuous variable have been grouped as a frequency or relative frequency histogram, the class with the highest frequency is called the modal class.

The midpoint of the modal class is taken as the mode.

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Figure 2.4(a)Relative frequency histograms for the milk data

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Figure 2.4(b)Relative frequency histograms for the GPA data

2.3 Measures of Variability

Variability or dispersion is a very important characteristic of data. See Figure 2.5 for examples of variability or dispersion of data.

Definition: The range, R, of a set of n measurements is defined as the difference between the largest and the smallest measurements.

See Figure 2.6 for an example of two relative frequency distributions that have the same range but very different shape and variability.

Figure 2.7 shows the deviations of points from the mean. Table 2.1 shows a computation of for the data in Figure 2.7.

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2)( xx i

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Figure 2.5(a)Variability or dispersion of data

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Figure 2.5(b)Variability or dispersion of data

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Figure 2.6(a)Distributions with equal range and unequal variability

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Figure 2.6(b)Distributions with equal range and unequal variability

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Figure 2.7 Showing the deviations of points from the mean

Definition: The variance of a population of N measurements is defined to be the average of the squares of the deviations of the measurements about their mean .

The population variance is denoted by 2 and is given by the formula

This measure will be relatively large for highly variable data and relatively small for less variable data.

Definition: The variance of a sample of n measurements is defined to be the sum of the squared deviations of the measurements about their mean divided by (n 1).

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Nx i

22 )(

x

The sample variance is denoted by s 2 and is given by the

formula:

Definition: The standard deviation of a set of measurements is equal to the positive square root of the variance.

Notation:

n: number of measurements in the sample

s 2: sample variance

: sample standard deviation

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1)( 2

2

nxx

s i

2ss

The shortcut method for calculating s 2 :

where sum of the squares of the individual

measurements

and square of the sum of the individual

measurements.

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1

)( 22

2

nn

xx

s

ii

ix2

2)(i

x

. Points to remember about variance and standard deviation:

- The value of s is always greater than or equal to zero.

- The larger the value of s 2 or s, the greater the variability of

the data set.

- If s 2 or s is equal to zero, all measurements must have

the same value.

- The standard deviation s is computed in order to have a measure of variability measured in the same units as the observations.

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Empirical Rule: Given a distribution of measurements that is

approximately mound-shaped:

- The interval (s) contains approximately 68% of the measurements

- The interval (2s) contains approximately 95% of the measurements.

- The interval (3s) contains almost all of the measurements.

The Empirical Rule applies to data with a normal distribution and many other types of data.

Use the Empirical Rule when the data distribution is roughly mound-shaped.

©1998 Brooks/Cole Publishing/ITP

2.6 Measures of Relative Standing

Definition: The sample z score is a measure of relative standing

defined by

A z-score measures the distance between an observation and the mean, measured in units of standard deviation.

An outlier is an unusually large or small observation.

z-scores between 2 and 2 are highly likely.

Z-scores exceeding 3 in absolute value are very unlikely.

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s

xxz

score-

Definition: A set of n measurements on the variable x has been arranged in order of magnitude.The pth percentile is the value of x that exceeds p% of the measurements and is less than the remaining (100 p)%.

Example 2.13

Suppose you have been notified that your score of 610 on the Verbal Graduate Record Examination placed you at the 60th percentile in the distribution of scores. Where does your score of 610 stand in relation to the scores of others who took the examination?

Solution

Scoring at the 60th percentile means that 60% of all examina-tion scores were lower than yours and 40% were higher.

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The median is the same as the 50th percentile. The 25th and 75th percentiles are called the lower and

upper quartiles.

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Figure 2.12

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Figure 2.13

Definition: A set of n measurements on the variable x has been arranged in order of magnitude.

The lower quartile (first quartile), Q1, is the value of x that

exceeds one-fourth of the measurements and is less than the

remaining 3/4. The second quartile is the median.

The upper quartile (third quartile), Q 3, is the value of x that

exceeds three-fourths of the measurements and is less than

one-fourth.

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When the measurements are arranged in order of magnitude, the lower quartile, Q1, is the value of x in the position .25(n 1).

The upper quartile, Q 3, is the value of x in the position

.75(n 1).

When these positions are not integers, the quartiles are found by interpolation, using the values in the two adjacent positions.

Definition: The interquartile range (IQR) for a set of measurements is the difference between the upper and lower quartiles; that is, IQR Q 3 Q 1.

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2.7 The Box Plot

From a box plot, you can quickly detect any skewness in the shape of the distribution and see whether there are any outliers in the data set.

To construct a box plot:

1. Calculate the median, the upper and lower quartiles, and the IQR.

2. Draw a horizontal line representing the scale of measurement.

3. Form a box above the line with the ends at Q 1 and Q 3 .

4. Draw a vertical line through the box at the location of m,

the median.

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Calculate lower and upper fences as follows:

Inner fences: Q 1 1.5(IQR) and Q 3 1.5(IQR)

- Measurements outside the lower and upper fences are called suspect outliers.

- Whiskers extend to the largest and smallest measurements inside the fences.

To finish the box plot:

- Locate the largest and smallest values using the scale along

the horizontal axis, and connect them to the box with horizontal lines called whiskers.

- Any suspect outliers are marked with an asterisk (*).

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Figure 2.15 shows the various values associated with the box plot. Example 2.15 exhibits the calculations for and the plotting of a box plot.

Figure 2.15

Skewed distributions usually have a long whisker in the direction of the skewness, and the median line is drawn away from the direction of the skewness.

©1998 Brooks/Cole Publishing/ITP