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**Describing Data Using Numerical Measures**

Chapter 3 Describing Data Using Numerical Measures

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**Chapter 3 - Chapter Outcomes**

After studying the material in this chapter, you should be able to: • Compute the mean, median, and mode for a set of data and understand what these values represent. • Compute the range, variance, and standard deviation and know what these values mean. • Know how to construct a box and whiskers graph and be able to interpret it.

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**Chapter 3 - Chapter Outcomes (continued)**

After studying the material in this chapter, you should be able to: • Compute the coefficient of variation and z scores and understand how they are applied in decision-making situations. • Use numerical measures along with graphs, charts, and tables to effectively describe data.

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**Parameters and Statistics**

A parameter is a measure computed from the entire population. As long as the population does not change, the value of the parameter will not change.

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**Parameters and Statistics**

A statistic is a measure computed from a sample that has been selected from the population. The value of the statistic will depend on which sample is selected.

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Mean The mean is a numerical measure of the center of a set of quantitative measures computed by dividing the sum of the values by the number of values in the data set.

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**N = number of data values xi = ith individual value of variable x**

Population Mean where: = population mean (mu) N = number of data values xi = ith individual value of variable x

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**Population Mean Example 3-1**

Table 3-1: Foster City Hotel Data

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**Population Mean Example 3-1**

The population mean for the number of rooms rented is computed as follows:

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**Sample Mean where: = sample mean (pronounced “x-bar”) n = sample size**

xi = ith individual value of variable x

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**Sample Mean Housing Prices Example**

{xi} = {house prices} = {$144,000; 98,000; 204,000; ,000; 155,000; 316,000; 100,000}

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Median The median is the center value that divides data that have been arranged in numerical order (i.e. an ordered array) into two halves.

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**Median Housing Prices Example**

{xi} = {house prices} = {$144,000; 98,000; 204,000; ,000; 155,000; 316,000; 100,000} Ordered array: $98,000; 100,000; 144,000; 155,000; 177,000; 204,000; 316,000 Middle Value Median = 155,000

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**Median Another Housing Prices Example**

{xi} = {house prices} = {$144,000; 98,000; 204,000; ,000; 155,000; 316,000; 100,000; ,000; 177,000; 170,000} Ordered array: $98,000; 100,000; 144,000; 155,000; 170,000; 177,000; 177,000; 177,000; 204,000; 316,000 Middle Values Median = (170, ,000)/2 = 173,500

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Skewed Data • Right-skewed data: Data are right-skewed if the mean for the data is larger than the median. • Left-skewed data: Data are left -skewed if the mean for the data is smaller than the median.

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**Skewed Data (Figure 3-3) a) Right-Skewed b) Left-Skewed c) Symmetric**

Median Mean Mean Median Mean = Median a) Right-Skewed b) Left-Skewed c) Symmetric

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Mode The mode is the value in a data set that occurs most frequently. A data set may have more than one mode if two or more values tie for the highest frequency. A data set might not have a mode at all if no value occurs more than one time.

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**Mode Housing Prices Example**

{xi} = {house prices} = {$144,000; 98,000; 204,000; ,000; 155,000; 316,000; 100,000; ,000; 177,000; 170,000} Data array: $98,000; 100,000; 144,000; 155,000; 170,000; 177,000; 177,000; 177,000; 204,000; 316,000 Mean = 1,718,000/10 = 171,800 Median = 173,500 Mode = 177,000

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**The median is the 50th percentile.**

Percentiles The pth percentile in a data array is a value that divides the data into two parts. The lower segment contains at least p% and the upper segment contains at least (100 - p)% of the data. The median is the 50th percentile.

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**The median corresponds to the second quartile.**

Quartiles Quartiles in a data array are those values that divide the data set into four equal-sized groups. The median corresponds to the second quartile.

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Measures of Variation A set of data exhibits variation if all of the data are not the same value.

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**R = Maximum Value - Minimum Value**

Range The range is a measure of variation that is computed by finding the difference between the maximum and minimum values in the data set. R = Maximum Value - Minimum Value

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**Interquartile Range = Third Quartile - First Quartile**

The interquartile range is a measure of variation that is determined by computing the difference between the first and third quartiles. Interquartile Range = Third Quartile - First Quartile

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**Variance & Standard Deviation**

The population variance is the average of the squared distances of the data values from the population mean. The standard deviation is the positive square root of the variance.

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**2 = population variance (sigma squared)**

where: = population mean N = population size 2 = population variance (sigma squared)

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**Population Variance (Bryce Lumber Example)**

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**Population Standard Deviation (Bryce Lumber Example)**

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**where: = sample mean n = sample size s2 = sample variance**

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**Sample Standard Deviation**

where: = sample mean n = sample size s = sample standard deviation

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**Coefficient of Variation**

The coefficient of variation is the ratio of the standard deviation to the mean expressed as a percentage. The coefficient of variation is used to measure the relative variation in the data.

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**Coefficient of Variation**

Population Coefficient of Variation Sample Coefficient of Variation

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The Empirical Rule If the data distribution is bell-shaped, then the interval: contains approximately 68% of the values in the population or the sample contains approximately 95% of the values in the population or the sample contains virtually all of the data values in the population or the sample

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**The Empirical Rule (Figure 3-11)**

95% 68% X

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**Tchebysheff’s Theorem**

Regardless of how the data are distributed, at least (1 - 1/k2) of the values will fall within k standard deviations of the mean. For example: At least (1 - 1/12) = 0% of the values will fall within k=1 standard deviation of the mean At least (1 - 1/22) = 3/4 = 75% of the values will fall within k=2 standard deviation of the mean At least (1 - 1/32) = 8/9 = 89% of the values will fall within k=3 standard deviation of the mean

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**Standardized Data Values**

A standardized data value refers to the number of standard deviations a value is from the mean. The standardized data values are sometimes referred to as z-scores.

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**Standardized Data Values**

STANDARDIZED POPULATION DATA where: x = original data value = population mean = population standard deviation z = standard score (number of standard deviations x is from )

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**Standardized Data Values**

STANDARDIZED SAMPLE DATA where: x = original data value = sample mean s = sample standard deviation z = standard score

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**Key Terms • Coefficient of Variation • Data Array • Empirical Rule**

• Interquartile Range • Left-Skewed Data • Mean • Median • Parameter • Percentiles • Quartiles • Range • Right-Skewed Data • Skewed Data • Standard Deviation • Standardized Data Values • Statistic

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**Key Terms (continued) • Symmetric Data • Tchebysheff’s Theorem**

• Variance • Variation

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