BUS 220:Elementary Statistics

Chapter 3: Numerical Measures

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Chapter 3: GOALS

1. Calculate the measures of central tendency including arithmetic mean, weighted mean, median, mode, and geometric mean.

2. Understand the advantages and disadvantages of those measures.

3. Identify whether a distribution is skewed or symmetric by looking at the positions of mean, median and mode.

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Chapter 3: GOALS…

4. Calculate the measures of dispersion including range, mean deviation, variance, and standard deviation.

5. Understand the characteristics, advantages, and disadvantages of those measures.

6. Understand the Chebshev’s theorem and the empirical rule as they are applied to a set of observation.

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Numerical Descriptive MeasuresMeasures of

central tendency Arithmetic Mean Weighted Mean Median Mode Geometric Mean

Measures of Dispersion

Range Mean Deviation Variance Standard

Deviation

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Arithmetic MeanFor ungrouped data, the population/sample arithmetic mean is the sum of all the population/sample values divided by the total number of values:

population mean sample mean

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EXAMPLE – Population Mean

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EXAMPLE – Sample Mean

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Properties of the Arithmetic Mean

1. Every set of interval-level and ratio-level data has a mean.

2. All the values are included in computing the mean.

3. The mean is unique.4. The sum of the deviations of each value from

the mean is zero.

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Weighted Mean The weighted mean of a set of

numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula:

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EXAMPLE – Weighted Mean

The Carter Construction Company pays its hourly employees $16.50, $19.00, or $25.00 per hour. There are 26 hourly employees, 14 of which are paid at the $16.50 rate, 10 at the $19.00 rate, and 2 at the $25.00 rate. What is the mean hourly rate paid the 26 employees?

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The Median

The Median is the midpoint of the values after they have been ordered from the smallest to the largest.

There are as many values above the median as below it in the data array.

For an even set of values, the median will be the arithmetic average of the two middle numbers.

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EXAMPLES - MedianThe ages for a sample of

five college students are:

21, 25, 19, 20, 22

Arranging the data in ascending order gives:

19, 20, 21, 22, 25.

Thus the median is 21.

The heights of four basketball players, in inches, are:

76, 73, 80, 75

Arranging the data in ascending order gives:

73, 75, 76, 80.

Thus the median is 75.5

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Properties of the Median

1. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur.

2. It can be computed for ratio-level, interval-level, and ordinal-level data.

3. It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class.

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The Mode The mode is the value of the

observation that appears most frequently.

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Example - Mode

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The Geometric Mean Useful in finding the average change of percentages,

ratios, indexes, or growth rates over time. Has a wide application in business and economics in

finding the percentage changes in sales, salaries, or economic figures, such as the GDP and interest rate which compound or build on each other.

Will always be less than or equal to the arithmetic mean.

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EXAMPLE – Geometric Mean

The return on investment earned by Atkins construction Company for four successive years was: 30, 20, -40, and 200 percent.

The geometric mean rate of return on investment is:

..).)(.)(.)(.(GM 2941808203602131 44

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Dispersion: Why Study Dispersion? Mean or the median, only describes the center of the

data, but it does not tell us anything about the spread of the data.

For example, if your nature guide told you that the river ahead averaged 3 feet in depth, would you want to wade across on foot without additional information? Probably not. You would want to know something about the variation in the depth.

A second reason for studying the dispersion in a set of data is to compare the spread in two or more distributions.

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Samples of Dispersions

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Measures of Dispersion

Range

Mean Deviation

Variance and Standard Deviation

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EXAMPLE – Range

The number of cappuccinos sold at the Starbucks location in the Orange Country Airport between 4 and 7 p.m. for a sample of 5 days last year were 20, 40, 50, 60, and 80. Determine the range for the number of cappuccinos sold.

Range = Largest – Smallest value = 80 – 20 = 60

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EXAMPLE – Mean Deviation

Determine the mean deviation for the number of cappuccinos sold.

Exercise 38—return on investment of 8 companies 10.6 12.6 14.8 18.2 12.0 14.8 12.2 15.6

Range 18.2-10.6=7.6% Arithmetic mean 13.85% Mean deviation 2.0% Interpret the results: The ROI or

the eight companies ranges from 10.6 to 18.2 which is 7.6% in range. On the average the return on investment deviates 2% from the mean of 13.85%.

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EXAMPLE – Variance and Standard Deviation

The number of traffic citations issued during the last five months in Beaufort County, South Carolina, are 38, 26, 13, 41, and 22. What is the population variance?

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EXAMPLE – Sample Variance

The hourly wages for a sample of part-time employees at Home Depot are: $12, $20, $16, $18, and $19. What is the sample variance?

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Using Excel for Descriptive StatisticsTable 2–4 in Chapter 2 shows the prices of the 80 vehicles sold last

month at Whitner Autoplex in Raytown, Missouri. Determine the mean and the median selling price.

The mean and the median selling prices are reported in the following Excel output. There are 80 vehicles in the study. So the calculations with a calculator would be tedious and prone to error.

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Read the lecture note on how to activate the analysis ToolPak for Excel

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The Relative Positions of the Mean, Median and the Mode

Real Estate Data from Jacksonville, FL

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Real Estate Data from Jacksonville, FL

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Application of Standard Deviation Because it is standardized it is meaningful

to use standard deviation to compare between two samples

E.g. two samples of productions rate at two plants has approximately the same mean, how if one has std deviation of 10 and another is 7 we can conclude that the first plant has more disperse rate of productions

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Application of Standard Deviation: Chebyshev’s TheoremFor any set of observations (sample or population), the proportion (percentage) of the values that lie within k standard deviations of the mean is at least 1 – 1/k2 percent.

where k is any constant greater than 1

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Chebyshev’s Theorem

The arithmetic mean biweekly amount contributed by the Dupree Paint employees to the company’s profit-sharing plan is $51.54, and the standard deviation is $7.51. At least what percent of the distributions lie within plus 3.5 standard deviations and minus 3.5 standard deviations of the mean?

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Application of Standard Deviation: The Empirical Rule