Describing Data Measures of Location Goals Chapter Three Describing Data Measures of Location Goals When you have completed this chapter, you will be able to: Calculate the arithmetic mean, median, mode, wighted mean, and the geometric mean. 2. Explain the characteristics, uses, advantages, and disadvantages of each measure of location. 3. Identify the position of the arithmetic mean, median, and mode for both symmetric and skewed distributions.

Compute and interpret the range, the mean deviation, the variance, and the standard deviation. Explain the characteristics, uses, advantages, and disadvantages of each measure of dispersion. Understand Chebyshev’s theorem and the Normal, or Empirical, Rule as they relate to a set of observations. Compute and interpret quartiles, the interquartile range, and the coefficient of variation.

1-Introduction This chapter is concerned with two other numerical ways of describing data, namely, measures of central tendency and measures of dispersion, often called variation or the spread. Parameter A characteristic of a population. Statistic A characteristic of a sample.

2-Population mean

2-Population Mean Example Listed below are 12 automobile companies and the number of patents granted by the United States government to each last year

Number of Patents Granted 2-Population Mean Campany Number of Patents Granted General Motor 511 Nissan 385 Daimler-Chrysler 275 Toyota 257 Honda 249 Ford 234 Mazda 210 Audi 97 Porsche 50 Mitsubishi 36 Volvo 23 BMW 13

2-Population Mean The typical number of patents received by an automobile company is 195. Because we considered all the companies receiving patents, this value is a population parameter.

3-The Sample Mean

3-The Sample Mean Example The Merrill Lynch Global Fund specializes in long-term obligations of foreign countries. We are interested in the interest rate on these obligations. A random sample of six bonds revealed the following.

3-The Sample Mean Issue Interest Rate Australian Gov’t Bonds 9.50% Belgian Gov’t bonds 7.25 Canadian gov’t bonds 6.50 French Gov’t “B-TAN” 4.75 Italian gov’t bonds 12.00 Spanish gov’t bonds 8.30

3-The Sample Mean The arithmetic mean interest rate of the sample of the long-term obligations is 8.05 percent.

4-The Properties of the Arithmetic Mean Every set of interval-level and ratio-level data has a mean. All the values are included in computing the mean. A set of data has only one mean. The mean is unique. Deviation from the mean sum to zero Mean is unduly affected by unsusually large and small values.

5-Weighted Mean

5-Weighted Mean Example The Carter Construction Company pays its hourly employees $6.50, $7.50, or $8.50 per hour. There are 26 hourly employees, 14 are paid at rate $6.50, 10 at the $7.50 rate, and 2 at the $8.50 rate. What is the weighted mean hourly rate paid the 26 employees?

6-The Median Median The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to the smallest. Fifty percent of the observations are above the median and fifty percent below the median

6-The Median Example The number of rooms in the seven hotels in downtown Pittsburgh is 713, 300, 618, 595, 311, 401, and 292. Find the median. Solution Step1 Arrange the data in order 292, 300, 311, 401, 595, 618, 713 Step2 Select the middle value Hence, the median is 401 rooms.

6-The Median If the number of observations in the data is even the median of the data is the arithmetic mean of the two middle values. Example: The number of cloudy days for the top ten cloudiest cities is shown. Find the median. 209, 223, 211, 227, 213, 240, 240, 211, 229, 212

6-The Median Solution: Arrange the data in order 209, 211, 211, 212, 213, 223, 227, 229, 240, 240

6-The Mode Mode The value of observation that appears most frequently. Mode is especially useful in describing nominal and ordinal levels of easurement. Example The following data represent the duration (in days) of U.S. space shuttle voyages for the years 1992 – 1994. Find the mode. 8, 9, 9, 14, 8, 8, 10, 7, 6, 9, 7, 8, 10, 14, 11, 8, 14, 11

6-The Mode Solution It is helpful to arrange the data in order, although it is not necessary 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 14, 14, 14 Since 8-day voyages occurred 5 times, a frequency larger than any other number, the mode for the data set is 8.

6-The Mode Example Find the mode for the number of coal employees per county for ten selected counties in southwestern Pennsylvania. 110, 731, 1031, 84, 20, 118, 1162, 1977, 103, 752 Solution Since each value occurs only once, there is no mode.

6-The Mode Example Eleven different automobiles were listed at a speed of 15 miles per hour for stopping distances. The data are shown below. Find the mode. 15, 18, 18, 18, 20, 22, 24, 24, 24, 26, 26 Solution Since 18 and 24 both occur 3 times, the modes are 18 and 24. This data set is said to be bimodal.

6-The Geometric Mean The geometric mean of a set of n positive numbers is defined as the nth root of the product of n values.

6-The Geometric Mean Another application of the geometric mean is to find an average percent increase over a period of time.

6-The Geometric Mean Example The population of a community in 1988 was 2 persons, by 1998 it was 22. What is the average annual rate of percentage increase during the period?

7- The Mean, Median, and Mode of Grouped Data Arithmetic Mean of Grouped Data X is the mid-value, or midpoint, of each class f is the frequency in each class n is the total number of frequency

7- The Mean, Median, and Mode of Grouped Data Example Below is a frequency table showing the distribution of grade points of students. Find the meian. Class Frequency 30 up to 40 4 40 up to 50 6 50 up to 60 8 60 up to 70 12 70 up to 80 9 80 up to 90 7 90 up to 100

7- The Mean, Median, and Mode of Grouped Data Class Freq Midpoint fX 30 up to 40 4 35 140 40 up to 50 6 45 270 50 up to 60 8 55 440 60 up to 70 12 65 780 70 up to 80 9 75 675 80 up to 90 7 85 595 90 up to 100 95 380

7- The Mean, Median, and Mode of Grouped Data Median of Grouped Data L is the lower limit of the class containing the median n is the total number of frequencies f is the frequency in the median class CF is the cumulative number of frequencies in all the classes preceding the class containing the median. i is the width of the class in which median lies.

7- The Mean, Median, and Mode of Grouped Data Example The data involving the selling prices of vehicles at Whitner Pontiac are shown in the table below. What is the median selling price for a new vehicle sold by Whitner Pontiac?

7- The Mean, Median, and Mode of Grouped Data Selling Price ($ thousands) Number Sold ( f ) CF 12 up to 15 8 15 up to 18 23 31 18 up to 21 17 48 21 up to 24 18 66 24 up to 27 74 27 up to 30 4 78 30 up to 33 1 79 33 up to 36 80 Total

7- The Mean, Median, and Mode of Grouped Data

7- The Mean, Median, and Mode of Grouped Data Example A sample of the daily production of transceivers at Scott Electronics was organized into the following distribution. Estimate the median daily production. (Answer: 105.5) Daily Production Frequency 80 up to 90 5 90 up to 100 9 100 up to 110 20 110 up to 120 8 120 up to 130 6 130 up to 140 2

7- The Mean, Median, and Mode of Grouped Data For data grouped into a frequency distribution, the mode can be approximated by the midpoint of the class containing the largest number of class frequencies.

THE END! 8-Selecting an Average for Data in a Frequency Distribution The mean is influenced more than the median or mode by a few extremely high or low values. If the distribution is highly skewed, the mean would not be a good average to use. The median and mode would be more representative. THE END!