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Describing Data: Numerical Measures

Describing Data: Numerical Measures. Chapter 3. GOALS. Calculate the arithmetic mean, weighted mean, median, mode, and geometric mean. Explain the characteristics, uses, advantages, and disadvantages of each measure of location.

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Describing Data: Numerical Measures

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  1. Describing Data:Numerical Measures Chapter 3

  2. GOALS • Calculate the arithmetic mean, weighted mean, median, mode, and geometric mean. • Explain the characteristics, uses, advantages, and disadvantages of each measure of location. • Identify the position of the mean, median, and mode for both symmetric and skewed distributions. • Compute and interpret the range, mean deviation, variance, and standard deviation. • Understand the characteristics, uses, advantages, and disadvantages of each measure of dispersion. • Understand Chebyshev’s theorem and the Empirical Rule as they relate to a set of observations.

  3. Characteristics of the Mean The arithmetic mean is the most widely used measure of location. It requires the interval scale. Its major characteristics are: • All values are used. • It is unique. • The sum of the deviations from the mean is 0. • It is calculated by summing the values and dividing by the number of values.

  4. Population Mean For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values:

  5. Population MeanFor ungrouped data (data not in a frequency distribution)

  6. A Parameteris a measurable characteristic of a population. 56,000 The Kiers family owns four cars. The following is the current mileage on each of the four cars. The mean mileage for the cars is 40,333 1/3 miles. 42,000 23,000 73,000

  7. EXAMPLE – Population Mean

  8. Sample Mean • For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values:

  9. Sample MeanFor ungrouped data (data not in a frequency distribution)

  10. EXAMPLE – Sample Mean

  11. A statisticis a measurable characteristic of a sample. A sample of five executives received the following bonus last year ($000): 14.0, 15.0, 17.0, 16.0, 15.0 Find The Sample Mean:

  12. 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 a unique mean. • The mean is affected by unusually large or small data values. • The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero.

  13. Consider the set of values: 3, 8, and 4. The mean is 5. Later, this will become relevant information when we calculate the variation and standard deviation (This is the reason we will have to square the deviations). Sum Of The Deviations Of Each Value From The Mean Is Zero

  14. Weighted Mean • The weighted meanof a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula:

  15. Weighted Mean • Instead of adding all the observations, we multiply each observation by the number of times it happens. • The weighted mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula: or

  16. 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?

  17. The Median • The Medianis themidpoint 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 odd set of values, the median will be the middle number. • For an even set of values, the median will be the arithmetic average of the two middle numbers. • Median is the measure of central tendency usually used by real estate agents. Why?...

  18. Median Example 2nd example:

  19. Median Example in Excel

  20. Properties of the Median • There is a unique median for each data set. • It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. • It can be computed for ratio-level, interval-level, and ordinal-level data. • It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class.

  21. EXAMPLES - Median 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 The 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.

  22. Mode • The Mode is the value of the observation that appears most frequently. • The mode is especially useful in describing nominal and ordinal levels of measurement. • There can be more than one mode.

  23. Mode Example – Nominal Level Data

  24. Mode Example – Nominal Level Data • With Nominal Data you would count to see which occurs most frequently • You can build a Frequency Table using the COUNTIF function in Excel. • The one that occurs the most is the Mode More data…

  25. Example – Mode – Ratio Level Data

  26. Mean, Median, Mode Using Excel Table 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.

  27. Mean, Median, Mode Using “Descriptive Statistics” in Excel (From Textbook)

  28. The Relative Positions of the Mean, Median and the Mode

  29. The Geometric Mean • Useful in finding the average change of percentages, ratios, indexes, or growth rates over time. • It has a wide application in business and economics because we are often interested in finding the percentage changes in sales, salaries, or economic figures, such as the GDP, which compound or build on each other. • The geometric mean will always be less than or equal to the arithmetic mean. • The GM gives a more conservative figure that is not drawn up by large values in the set. • The geometric mean of a set of n positive numbers is defined as the nth root of the product of n values. • The formula for the geometric mean is written:

  30. Geometric Mean • The GM of a set of n positive numbers is defined as the nth root of the product of n values. The formula is either (both are true):

  31. Geometric Mean Example 1:Percentage Increase

  32. Verify Geometric Mean Example The GM gives a more conservative figure that is not drawn up by large values in the set.

  33. EXAMPLE – Geometric Mean (2) The return on investment earned by Atkins construction Company for four successive years was: 30 percent, 20 percent, -40 percent, and 200 percent. What is the geometric mean rate of return on investment?

  34. Another Use Of GM:Ave. % Increase Over Time • Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another • Where n = number of periods

  35. Example for GM: Ave. % Increase Over Time • The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in 2000. That is, the geometric mean rate of increase is 1.27%. • The annual rate of increase is 1.27% • For the years 1992 through 2000, the rate of female enrollment growth at American colleges was 1.27% per year

  36. Dispersion Why Study Dispersion? • A measure of location, such as the mean or the median, only describes the center of the data. It is valuable from that standpoint, 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.

  37. Dictionary Definitions:Dispersion, Variation, Deviation • Dispersion • The spatial property of being scattered about over an area or volume • The degree of scatter of data, usually about an average value, such as the median or mean • Variation • The act of changing or altering something slightly but noticeably from the norm or standard • Deviation • A variation that deviates from the standard or norm

  38. Dispersion • How spread out is the data? • What is the average of all the deviations? • A small value for a measure of dispersion indicates that the data are clustered around the typical value (mean) • Mean can fairly represent the data • A large value for a measure of dispersion indicates that the data are not clustered around the typical value (mean) • Mean may not fairly represent the data

  39. Samples of Dispersions

  40. Measures of Dispersion • Range • Mean Deviation • Variance and Standard Deviation

  41. Range • Range = Highest Value – Lowest Value • Excel = MAX – MIN functions • The difference between the largest and the smallest value • Advantage: • It is easy to compute and understand • Disadvantage: • Only two values are used in its calculation • It is influenced by an extreme value

  42. Range Example

  43. Mean Deviation measures the mean amount by which the values in a population, or sample, vary from their mean Mean Deviation (MA or MAD)

  44. Variance In Excel: 1) Population Variance  VARP function 2) Sample Variance  VAR function

  45. Standard Deviation In Excel: 1) Population Variance  STDEVP function 2) Sample Variance  STDEV function The primary use of the statistic s2 is to estimate σ2, therefore (n-1) is necessary to get a better representation of the deviation or dispersion in the data (n tends to underestimate)

  46. 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 mean deviation for the number of cappuccinos sold. Range = Largest – Smallest value = 80 – 20 = 60

  47. EXAMPLE – Mean Deviation 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 mean deviation for the number of cappuccinos sold.

  48. EXAMPLE – Variance and Standard Deviation The number of traffic citations issued during the last five months in Beaufort County, South Carolina, is 38, 26, 13, 41, and 22. What is the population variance? 2 ^(1/2) =  = 10.22441

  49. EXAMPLE – Sample Variance and Sample Standard Deviation 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 and sample standard deviation? s2 ^(1/2) = s = 3.162278

  50. EXAMPLE – Sample Standard Deviation (p1)

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