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Describing Data: Numerical Measures Chapter 3 GOALS 1. Calculate the arithmetic mean, weighted mean, median, mode, and geometric mean. 2. Explain the characteristics, uses, advantages, and disadvantages of each measure of location. 3. Identify the position of the mean, median, and mode for both symmetric and skewed distributions. 4. Compute and interpret the range, mean deviation, variance, and standard deviation. 5. Understand the characteristics, uses, advantages, and disadvantages of each measure of dispersion. 6. Understand Chebyshev’s theorem and the Empirical Rule as they relate to a set of observations. 2 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. 3 Population Mean For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values: 4 Population Mean For ungrouped data (data not in a frequency distribution) Sum of all the values in the population Population Mean = Number of values in the population X N Define Variables & Symbols µ = The population mean = “mu” N = Total number of observations X = A particular value Σ = Indicates the operation of adding = “sigma” ΣX = Sum of the X values 5 A Parameter is a measurable characteristic of a population. The Kiers 56,000 family owns 42,000 four cars. The 23,000 following is the current mileage 73,000 on each of the four cars. The mean mileage for the cars is 40,333 1/3 miles. 6 EXAMPLE – Population Mean 7 Sample Mean For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values: 8 Sample Mean For ungrouped data (data not in a frequency distribution) Sum of all the values in the sample Sample Mean = Number of values in the sample X X n Define Variables & Symbols X X = Sample Mean = "X bar" n = Total number of observations X = A particular value Σ = Indicates the operation of adding = “sigma” ΣX = Sum of the X values 9 EXAMPLE – Sample Mean 10 A statistic is a measurable characteristic of a sample. A sample of five 14.0, executives 15.0, received the 17.0, following 16.0, bonus last year ($000): 15.0 Find The Sample Mean: 11 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. 12 Sum Of The Deviations Of Each Value From The Mean Is Zero Deviation ( X X ) X A particular value, X Mean Consider the set of values: 3, 8, and 4. The mean is 5. ( X X ) (3 5) (8 5) (4 5) 0 Later, this will become relevant information when we calculate the variation and 13 standard deviation (This is the reason we will have to square the deviations). 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: 14 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: Xw (w 1X1 w 2 X 2 ... w n X n ) Xw (wX) w or (w 1 w 2 ...w n ) Define Variables & Symbols X w = Weighted Mean = "X bar sub w" X1 = A particular value X2 = A particular value w1 = A particular weight w2 = A particular weight 15 Σ = Indicates the operation of adding = “sigma” 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? 16 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 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?... 17 Median Example $80,000.00 $70,000.00 $275,000.00 $65,000.00 $60,000.00 Prices Ordered Prices Ordered from Low to High from High to Low $60,000 $275,000 65,000 80,000 70,000 ← Median → 70,000 2nd 80,000 65,000 example: 275,000 60,000 Prices Ordered Prices Ordered from Low to High from High to Low $60,000 $275,000 65,000 80,000 70,000 (70000 + 75000) 75,000 = $72,500 75,000 2 70,000 80,000 65,000 275,000 60,000 18 Median Example in Excel 19 Properties of the Median There is a unique median for each data set. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. It can be computed for ratio-level, interval- level, and ordinal-level data. It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class. 20 EXAMPLES - Median The ages for a sample of The heights of four five college students basketball players, in are: inches, are: 21, 25, 19, 20, 22 76, 73, 80, 75 Arranging the data in Arranging the data in ascending order gives: ascending order gives: 73, 75, 76, 80. 19, 20, 21, 22, 25. Thus the median is 75.5 Thus the median is 21. 21 Mode • The Mode is the value of the observation that appears most frequently. • The mode is especially useful in describing nominal and ordinal levels of measurement. • There can be more than one mode. 22 Mode Example – Nominal Level Data Company has developed five bath oils Company conducts marketing survey to determine which oil customers prefer Number of Respondents Favoring Various Bath Oils 400 358 Number of Respondents 300 200 193 110 100 92 43 0 Amor Lamoure Soothing Smell Nice Far Out 23 Mode Bath Oil Mode Example – Nominal Level Data • With Nominal Data Respondents Smell Nice Frequency Table you would count to Lamoure Smell Nice 92 =COUNTIF($A$2:$A$797,C3) see which occurs Lamoure Lamoure 358 =COUNTIF($A$2:$A$797,C4) Soothing Soothing 193 =COUNTIF($A$2:$A$797,C5) most frequently Smell Nice Amor 110 =COUNTIF($A$2:$A$797,C6) • You can build a Smell Nice Far Out 43 =COUNTIF($A$2:$A$797,C7) Soothing Frequency Table Amor Lamoure using the Soothing COUNTIF function Soothing Smell Nice in Excel. Smell Nice Lamoure • The one that Amor Lamoure occurs the most is the Mode More data… 24 Example – Mode – Ratio Level Data Salaries $ 35,000 $ 49,100 $ 60,000 $ 60,000 $ 40,000 $ 58,000 $ 60,000 $ 60,000 $ 40,000 $ 65,000 $ 55,000 $ 60,000 $ 71,400 $ 60,000 $ 55,000 In Excel: $ 60,000 =MODE(G2:G16) 25 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. 26 Mean, Median, Mode Using “Descriptive Statistics” in Excel (From Textbook) 27 The Relative Positions of the Mean, Median and the Mode 28 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: 29 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): GM % 1 n ( X 1)( X 2)( X 3)...(Xn) GM % n ( X 1)( X 2)( X 3)...(Xn) 1 Define Variables & Symbols GM = Geometric Mean X1 = A particular number (1 + %) X2 = A particular number (1 + %) n = Number of postive numbers in set 30 Geometric Mean Example 1: Percentage Increase Starting Salary $41,000.00 Increase in salary Year 1 5% Increase in salary Year 2 15% GM 2 (1.05)(1.15) 1.09886 In Excel: 1.05 * 1.15 = 1.2075 GM = 1.2075 ^ (1/2) - 1 = 9.886% 31 Verify Geometric Mean Example Verify 1: Raise 1 = $41,000.00 * 5% = $2,050.00 Raise 2 = 43,050.00 * 15% = 6,457.50 Total $8,507.50 Verify 2: Raise 1 = $41,000.00 * 0.09886 = $4,053.39 Raise 2 = 45,053.39 * 0.09886 = 4,454.11 Total $8,507.50 The GM gives a more If We used Arithmetic Mean (5%+15%)/2 = 10% conservative figure that is not drawn Raise 1 = $41,000.00 * 10% = $4,100.00 up by large values Raise 2 = 45,100.00 * 10% = 4,510.00 in the set. Total $8,610.00 32 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? GM 4 (1.3 )(1.2 )( 0.6 )( 3.0 ) 4 2.808 1.294 33 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 (Value at end of all the periods) GM n 1 (Value at beginning of all the periods) 34 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%. 835,000 GM 8 1 .0127 755,000 •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 35 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. 36 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 Deviation ( X X ) 37 Dispersion 1. How spread out is the data? • What is the average of all the deviations? 2. 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 3. 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 38 Samples of Dispersions 39 Measures of Dispersion Range Mean Deviation Variance and Standard Deviation 40 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 41 Range Example Boomerangs made per day over ten day period Boomerangs made per day over ten day period ? ? ? ? ? ? ? ? ? ? 52 54 56 58 60 62 64 66 68 X Boomerangs made per day at Colorado Boomerangs ? ? ? ? ? ? ? ? ? ? 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 X Boomerangs made per day at Gel Boomerangs 42 Mean Deviation (MA or MAD) • Mean Deviation measures the mean amount by which the values in a population, or sample, vary from their mean X X MD n Define Variables & Symbols X X = Sample Mean = "X bar" n = Total number of observations X = A particular value Σ = Adding = “sigma” MD = Mean Deviation ? = Absolute Value (distance from zero) 43 Variance ( X ) 2 ( X X ) 2 2 s 2 N n 1 Define Variables & Symbols Define Variables & Symbols 2 = Pop. Variance = "sigma squared" s2 = Sample Variance Total number of observations n = (n as denominator tends to N = Total number of observations underestimate variance) Provides the appropriate correction n-1 = X = A particular value for underestimation X = A particular value μ = Pop. Mean X = Sample Mean In Excel: 1) Population Variance VARP function 2) Sample Variance VAR function 44 Standard Deviation ( X ) 2 ( X X )2 2 s s2 N n 1 Define Variables & Symbols Define Variables & Symbols 2 = Pop. Standard Deviation s s2 = Sample Standard Deviation Total number of observations n = (n as denominator tends to N = Total number of observations underestimate variance) Provides the appropriate correction n-1 = for underestimation X = A particular value X = A particular value μ = Pop. Mean X = Sample Mean 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 45 deviation or dispersion in the data (n tends to underestimate) 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 46 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. 47 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 48 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 49 EXAMPLE – Sample Standard Deviation (p1) 50 EXAMPLE – Sample Standard Deviation (p2) Conclusions The Variance is 14.48 square grams The sample standard deviation is 3.8 grams On average the weight of each box deviated from the sample mean by 3.8 grams. If the manufacturer of the cereal has 254 grams printed on the box, is the production efficient? Is their claim of 254 grams accurate or fair? If the historical sample standard deviation (s) is 2.7 grams, then an s = 3.8 indicates that it is reasonable to assume that the production is not efficient. Also, s = 2.7 grams would indicate that the claim on the box does not seem reasonable. 51 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 contributions lie within plus 3.5 standard deviations and minus 3.5 standard deviations of the mean? 52 Chebyshev’s Theorem • For any set of observations (sample or population & does not have to be normal distributed), the proportion of the values that lie within (+/-) k standard deviations of the mean is at least: 1 1 2) The smaller this 3) The bigger this 2 k 1) The bigger this • where k is any constant greater than 1 53 The Empirical Rule (also called Normal Rule) 54 34.13 34.13 % area % area 13.58 13.58 % area % area 2.15% 2.15% area area 0.13% 0.13% area area Xbar-3 Xbar-2 Xbar-1 Xbar Xbar +1 Xbar +2 Xbar +3 68.26% area 95.44% area 99.74% area 55 A sample of the rental rates at for 2 bedroom apartments in Seattle approximates a symmetrical, bell-shaped distribution. The sample mean is $925.00; The standard deviation is $110.00. Using the Empirical Rule, answer these questions: Sample Mean = Xbar = $925.00 Sample Standard Deviation = s = $110.00 Q1: About 68% of the rental rates are between what two amounts? Xbar +/- 1*s = $925.00 +/- 1*$110.00 ==> About 68% of the values lie between $815.00 and $1,035.00. Q2: About 95% of the rental rates are between what two amounts? Xbar +/- 2*s = $925.00 +/- 2*$110.00 ==> About 95% of the values lie between $705.00 and $1,145.00. Q3: Almost all of the rental rates are between what two amounts? Xbar +/- 3*s = $925.00 +/- 3*$110.00 ==> About Almost all of the values lie between $595.00 and $1,255.00. 56 The Arithmetic Mean of Grouped Data 57 The Arithmetic Mean of Grouped Data - Example Recall in Chapter 2, we constructed a frequency distribution for the vehicle selling prices. The information is repeated below. Determine the arithmetic mean vehicle selling price. 58 The Arithmetic Mean of Grouped Data - Example 59 Standard Deviation of Grouped Data 60 Standard Deviation of Grouped Data - Example Refer to the frequency distribution for the Whitner Autoplex data used earlier. Compute the standard deviation of the vehicle selling prices 61 Approximate Median from Frequency Distribution 1. Locate the class in which the median lies • Divide the total number of data values by 2. • Determine which class will contain this value. For example, if n=50, 50/2 = 25, then determine which class will contain the 25th value n 2. Use formula to arrive CF at estimate of Median: Median L 2 (i ) f Define Variables & Symbols L = Lower limit of the class containing the median n = Total number of frequencies f = Frequency in the median class Cumulative number of frequencies in all the classes CF = preceeding the class containing the medain i = width of the class in which the median lies 62 Approximate Median from Frequency Distribution Example 1 Selling Price Cumulative Total Number Frequencies 1st = 40 ($ Frequency Frequency 2 thousands) (f) CF 12 up to 15 8 8 40th value must be here 15 up to 18 23 31 18 up to 21 17 48 2nd L = 18 21 up to 24 18 66 n = 80 24 up to 27 8 74 f = 17 27 up to 30 4 78 CF = 31 30 up to 33 2 80 i = 3 Total 80 n 80 CF 31 Median L 2 (i ) 18 2 (3) 19.588235294 f 17 The estimated median vehicle selling price is $19,588.00 63 Selling Price Cumulative ($ Frequency Frequency thousands ) (f) CF 12 up to 15 8 8 15 up to 18 23 31 18 up to 21 17 48 21 up to 24 18 66 24 up to 27 8 74 27 up to 30 4 78 30 up to 33 2 80 Total 80 • Let’s think about this: – n/2 – CF = 80/2-31 = 9 vehicles to get from the 31st to the 40th – (n/2 – CF)/f = 9/17 = 9/17th of the distance through the class width of $3000 – ((n/2 – CF)/f)(i) = (9/17)($3000) = $1,588 – ((n/2 – CF)/f)(i) + L = $1,588 + $18,000 = $19,588, Median estimate 64 Estimate Range from Grouped Data Employee $ contributed to profit Frequency sharing plan f $50 Up To $55 4 55 Up To $60 7 60 Up To 65 9 65 Up To 70 22 70 Up To 75 40 75 Up To 80 24 80 Up To 85 15 85 Up To 90 9 Total 130 Upper Limit of Lower Limit of Estimated the Highest the Lowest Range = Class - Class Estimated Range = $90 - $50 = $40 65 End of Chapter 3 66