After completing this chapter, you should be able to:

Transcription

1 Chapter 2 Descriptive Statistics Chapter Goals After completing this chapter, you should be able to: Compute and interpret the mean, median, and mode for a set of data Find the range, variance, standard deviation, and coefficient of variation and know what these values mean Apply the empirical rule to describe the variation of population values around the mean Explain the weighted mean and when to use it Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-2 Figure 2.1: 1

2 2 CHAPTER 2. DESCRIPTIVE STATISTICS Chapter Topics Measures of central tendency, variation, and shape Mean, median, mode, geometric mean Quartiles Range, interquartile range, variance and standard deviation, coefficient of variation Symmetric and skewed distributions Population summary measures Mean, variance, and standard deviation Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-3 Figure 2.2:

3 2.1. MEASURE OF CENTRAL TENDENCY OR MEASURES OF LOCATION 3 Describing Data Numerically Describing Data Numerically Central Tendency Arithmetic Mean Median Mode Variation Range Interquartile Range Variance Standard Deviation Coefficient of Variation Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-5 Figure 2.3: We will consider the description of a data set,with and observations for the sample and population respectively Often referred to as data reduction 2.1 Measure of Central Tendency or Measures of Location

4 4 CHAPTER 2. DESCRIPTIVE STATISTICS Measures of Central Tendency Overview Central Tendency Mean Median Mode x n i1 n x Arithmetic average i Midpoint of ranked values Most frequently observed value Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-6 Figure 2.4: Population Mean = 1 X =1

5 2.1. MEASURE OF CENTRAL TENDENCY OR MEASURES OF LOCATION 5 Arithmetic Mean The arithmetic mean (mean) is the most common measure of central tendency For a population of N values: μ For a sample of size n: x N i1 n N n x x i x x 1 x x x N Sample size Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap n x i i1 1 2 Observed n values N Population values Population size Figure 2.5:

6 6 CHAPTER 2. DESCRIPTIVE STATISTICS Parameter: A characteristic of a population Usually we do not have the population available The Kiers family owns four vehicles. Following is the current mileage for each: 56,000; 23,000; 42,000; and 73,000. =( )4 = Sample Mean = 1 X =1 Statistic: A characteristic of a sample Sample mean is an estimator of the population mean.

7 2.1. MEASURE OF CENTRAL TENDENCY OR MEASURES OF LOCATION 7 Arithmetic Mean (continued) The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) Mean = Mean = Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-8 Figure 2.6:

8 8 CHAPTER 2. DESCRIPTIVE STATISTICS Notice that the unit for the sample mean is the same as the observations themselves. Mean is sensitive to Outliers! Combining Sample Means with different Sample Sizes: Weighted Means Suppose the sample mean in Economics 01 is 65 and that the mean male score is 62 and the female was 82. What is the proportion of female students in the class? and. We know that: = 1 = X =1 = 1 1 X + ( + ) =1 X =1 = ( + ) + If we let be the proportion of females then: ( + ) Solving for gives.15. =(1 ) + 65 = (1 ) Notice carefully to get the overall sum for the sample we must weight each of therespectivesamplemeans(maleandfemale)bytheirrelative proportion ( ) of the sample. Weighted mean:: = X =1 where X =1 Question: If given unemployment rates for each of the 10 provinces how would you calculate the unemployment rate for Canada?

9 2.1. MEASURE OF CENTRAL TENDENCY OR MEASURES OF LOCATION Grouped Data For grouped data we can calculate the sample mean as = 1 X =1 wherewehave classes and is the midpoint (class mark or midpoint) for class i with frequency. Approximations for Grouped Data Suppose a data set contains values m 1, m 2,..., m k, occurring with frequencies f 1, f 2,... f K For a population of N observations the mean is μ K i i1 f m For a sample of n observations, the mean is x N K i i1 n Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-43 i fm i where where N n K f i i1 K f i i1 Figure 2.7:

10 10 CHAPTER 2. DESCRIPTIVE STATISTICS Median-Md ( 2 ) The median Md of a set of data is a number such that half of the observations are below the number and half of the observations are above that number when the data are ordered in an array from lowest to highest. Median In an ordered list, the median is the middle number (50% above, 50% below) Median = 3 Median = 3 Not affected by extreme values Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-9 Figure 2.8:

11 2.1. MEASURE OF CENTRAL TENDENCY OR MEASURES OF LOCATION 11 Finding the Median The location of the median: Median position n 1 2 position in the ordered data If the number of values is odd, the median is the middle number If the number of values is even, the median is the average of the two middle numbers n 1 Note that is not the value of the median, only the 2 position of the median in the ranked data Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-10 Figure 2.9:

12 12 CHAPTER 2. DESCRIPTIVE STATISTICS Figure 2.10: Number of Movies Showing at Theater Movies Frequency Cumulative showing Frequency Median with Grouped Data For grouped data we get the rather complex equation (recall ogive approximation method) 2 = + where = lower boundary of class containing the median = width of class containing median = frequency of class containing median = minus the cumulative frequency of the class preceding the class 2 2 containing the median and being the total number of observations. The median class is 5-6, since it contains the 5th value (n/2 =5) Notes If n is odd then the median is the middle value of the ordered array. If n is even, then add the two central values and divide by two Suppose we have data = 10 (even) that has been ordered smallest to largest: Median= (10+15)/2=12.5. Median is not so sensitive to outliers in the data Unique median for each data set.

13 2.1. MEASURE OF CENTRAL TENDENCY OR MEASURES OF LOCATION Mode - Mo The mode Mo of a set of numbers most frequently occurring value of the data set. Ungrouped Data 24287modeis2 Grouped Data: The midpoint (class mark) of the class containing the largest number of class frequencies. Notice the mode need not be unique (i.e. bimodal) Not a good measure of central tendency Dependsonfrequencyofobservationratherthanabalanceoverallobservations. Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may may be no mode There may be several modes Mode = 9 No Mode Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-11 Figure 2.11:

14 14 CHAPTER 2. DESCRIPTIVE STATISTICS Relationship between Mean, Median and Mode Symmetric Distribution: Median=Mean=(Mode) Right Skewed Distribution (Positively skewed): Mean and Median are to the right of the Mode ModeMedianMean Left Skewed Distribution (Negatively Skewed): MeanandMedianaretotheleftoftheMode MeanMedianMode

15 2.2. MEASURES OF VARIABILITY OR DISPERSIONS Measures of Variability or Dispersions Measures of Variability Variation Range Interquartile Range Variance Standard Deviation Coefficient of Variation Measures of variation give information on the spread or variability of the data values. Same center, different variation Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-16 Figure 2.12:

16 16 CHAPTER 2. DESCRIPTIVE STATISTICS Range The range is the difference between the largest and the smallest values of the data. RANGE = A sample of five accounting graduates revealed the following starting salaries: $22,000, $28,000, $31,000, $23,000, $24,000. The range is $31,000 - $22,000 = $9,000. Problem with this measure of dispersion is that it depends upon only two numbers and is particularly sensitive to outliers in the data. Interquartile Range The interquartile range is 3-1, this is the range of the middle 50 % of the data. Interquartile Range Can eliminate some outlier problems by using the interquartile range Eliminate high- and low-valued observations and calculate the range of the middle 50% of the data Interquartile range = 3 rd quartile 1 st quartile IQR = Q 3 Q 1 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-19 Figure 2.13:

17 2.2. MEASURES OF VARIABILITY OR DISPERSIONS 17 Interquartile Range Example: X minimum Q1 Median (Q2) Q3 25% 25% 25% 25% X maximum Interquartile range = = 27 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-20 Figure 2.14:

18 18 CHAPTER 2. DESCRIPTIVE STATISTICS Quartiles Quartiles split the ranked data into 4 segments with an equal number of values per segment 25% 25% 25% 25% Q1 Q2 Q3 The first quartile, Q 1, is the value for which 25% of the observations are smaller and 75% are larger Q 2 is the same as the median (50% are smaller, 50% are larger) Only 25% of the observations are greater than the third quartile Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-21 Figure 2.15:

19 2.2. MEASURES OF VARIABILITY OR DISPERSIONS 19 Quartile Formulas Find a quartile by determining the value in the appropriate position in the ranked data, where First quartile position: Q 1 = 0.25(n+1) Second quartile position: Q 2 = 0.50(n+1) (the median position) Third quartile position: Q 3 = 0.75(n+1) where n is the number of observed values Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-22 Figure 2.16:

20 20 CHAPTER 2. DESCRIPTIVE STATISTICS Quartiles Example: Find the first quartile Sample Ranked Data: (n = 9) Q 1 = is in the 0.25(9+1) = 2.5 position of the ranked data so use the value half way between the 2 nd and 3 rd values, so Q 1 = 12.5 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-23 Figure 2.17: Deviation from the Mean The deviation from the sample mean is =1 We might consider one measure of variation as the Average Deviation: 1 X =1 [ ] except this is equal to 0 (recall proof in first chapter)

21 2.3. POPULATION VARIANCE Mean (Absolute) Deviations (MAD) = 1 X =1 Example: The weights of a sample of crates containing books for the bookstore are (in lbs.) 103, 97, 101, 106, 103. =12/5= Population Variance 2 = 1 = 1 X =1 X =1 [ ] Notes 2 0 Thesquarerootofthevariance, is called the population standard deviation or root-mean squared deviation of X. ismeasuredinthesameunitsasx. Sample Variance 2 = 1 P =1 [ 1 ] 2 Alternatively and try to derive this 2 = 1 1 [P =1 2 2 ] 2 = P =1 2 (P ) 2 1 Grouped data 2 = 1 1 X =1 [ ] 2

22 22 CHAPTER 2. DESCRIPTIVE STATISTICS Approximations for Grouped Data Suppose a data set contains values m 1, m 2,..., m k, occurring with frequencies f 1, f 2,... f K For a population of N observations the variance is σ 2 K i1 f (m μ) For a sample of n observations, the variance is s 2 K i i i1 Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-44 i N 2 f (m x) i n1 2 Figure 2.18: We can have two data sets with the same means but quite different sample variances. 2 is measured in 2 is measured in the same units as the observations themselves.

23 2.3. POPULATION VARIANCE 23 Calculation Example: Sample Standard Deviation Sample Data (x i ) : n = 8 Mean = x = 16 s (10 X ) 2 (12 x ) 2 (14 n 1 x ) 2 (24 x) 2 (10 16) 2 (12 16) 2 (14 16) (24 16) A measure of the average scatter around the mean Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-28 Figure 2.19:

24 24 CHAPTER 2. DESCRIPTIVE STATISTICS Measuring variation Small standard deviation Large standard deviation Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-29 Figure 2.20:

25 2.3. POPULATION VARIANCE 25 Empirical Rule for Bell Shaped (Normal) Distributions For data that give rise to histograms with the familiar (symmetric) bell-shaped appearance then: 68% of data lies within the interval ± 95% of data lies within the interval ± % of data lies within the interval ± 3

26 26 CHAPTER 2. DESCRIPTIVE STATISTICS Empirical Rule for Standard Deviations The Empirical Rule If the data distribution is bell-shaped, then the interval: μ 1σ contains about 68% of the values in the population or the sample 68% μ μ 1σ Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-34 Figure 2.21:

27 2.3. POPULATION VARIANCE 27 The Empirical Rule Normal μ 2σ contains about 95% of the values in the population or the sample μ 3σ contains about 99.7% of the values in the population or the sample 95% μ 2σ 99.7% μ 3σ Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-35 Figure 2.22:

28 28 CHAPTER 2. DESCRIPTIVE STATISTICS For most data, 90% or more of the values lie within plus or minus three standard deviations from the mean

29 2.4. COEFFICIENT OF VARIATION ( ) Coefficient of Variation ( ) = 100% for population = 100% for sample Adjusts the standard deviation (population or sample) by dividing by the mean again either by population or sample) Unit free and allows comparisons of variations of similar objects that control for size. Comparing Coefficient of Variation Stock A: Average price last year = $50 Standard deviation = $5 CV A Stock B: s x 100% Average price last year = $100 Standard deviation = $5 CV B $5 100% 10% $50 s $5 100% 100% 5% x $100 Both stocks have the same standard deviation, but stock B is less variable relative to its price Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-37 Figure 2.23:

30 30 CHAPTER 2. DESCRIPTIVE STATISTICS 2.5 Covariances: Population and Sample The Sample Covariance The covariance measures the strength of the linear relationship between two variables The population covariance: Cov (x,y) The sample covariance: xy Cov(x,y) s Only concerned with the strength of the relationship No causal effect is implied xy N i1 (x )(y ) n i1 i x N (x x)(y y) i n 1 i i y Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-45 Figure 2.24:

31 2.5. COVARIANCES: POPULATION AND SAMPLE 31 Interpreting Covariance Covariance between two variables: Cov(x,y) > 0 Cov(x,y) < 0 Cov(x,y) = 0 x and y tend to move in the same direction x and y tend to move in opposite directions x and y are independent Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-46 Figure 2.25:

32 32 CHAPTER 2. DESCRIPTIVE STATISTICS 2.6 Correlation: Population and Sample Coefficient of Correlation Measures the relative strength of the linear relationship between two variables Population correlation coefficient: Cov (x,y) ρ σ X σ Y Sample correlation coefficient: Cov(x, y) r s X s Y Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-47 Figure 2.26:

33 2.7. EXPLORATORY DATA ANALYSIS 33 Features of Correlation Coefficient, r Unit free Ranges between 1 and 1 The closer to 1, the stronger the negative linear relationship The closer to 1, the stronger the positive linear relationship The closer to 0, the weaker any positive linear relationship Statistics for Business and Economics, 6e 2007 Pearson Education, Inc. Chap 3-48 Figure 2.27: 2.7 Exploratory Data Analysis Some Remarks Often we want to display graphically a measure of central tendency and dispersion and the simplest way to do this is by a box plot. 2.8 BOX-and-Whisker PLOTS Plots5numbersalongasingleaxis:(Five Number Summary) Minimum and Maximum Values 1, 2 and 3 Example: Based on a sample of 20 deliveries, Marco s Pizza determined the following information: minimum value = 13 minutes, Q1 = 15 minutes, median = 18 minutes, Q3 = 22 minutes, maximum value = 30 minutes

34 34 CHAPTER 2. DESCRIPTIVE STATISTICS Figure 2.28: Vertical Box Plots Usually Box-and-Whisker Plots are horizontal, but for some reason many statistical programs (STATA and Excel) plot them vertically. 2.9 Mathematical Note: Linear Transformations Let and be two constants and =1 be a variable. An example would be with kilometers, and miles, then =0 and =06. A linear transformation is defined as: = + is said to be defined by a linear combination of. We note that is the intercept and is the slope of the equation.

35 2.10. STANDARDIZED VARIABLE ( ) 35 Notice that changes the origin and that changes the scale. How is the mean and variance of the variable Y related to the mean and the variance of X? Firstconsiderthepopulationcase: Mean of a Linear Transformation = 1 P =1 = 1 P =1 [ + ] = + 1 P =1 = + Variance of a Linear Transformation 2 = 1 = 1 = 1 P =1 [ ] 2 P =1 [ + { + }] 2 P =1 [{ }] 2 = 2 1 P =1 [ ] 2 = 2 2 This implies = It also follow that for the sample counterparts that = + 2 = 2 2 = 2.10 Standardized Variable ( ) Finally we may define a standardized variable (in terms of the sample) as = Notes on Standarizing

36 36 CHAPTER 2. DESCRIPTIVE STATISTICS This variable has a mean of zero (prove this using the results above on linear transformations) and a sample standard deviation and variance equal to 1 ( =, = 1 ). Variables defined in this way are sometimes referred to as, Z-scores. =0 2 = =1 We can do the same type of transformation in terms of the population magnitudes and. Notice that if an observation is below the mean then the is negative and above if it is positive. Questions There are many problems on means, variances and so on in LMM and you should make sure you are comfortable with the topics by doings as many as you can. NCT 2.28 a-c, 2.21, 2.29a,b,d, Linear Transformations Show the relationship between centigrade and Fahrenheit. What temperature measure shows greater variation and hence provides more detail on temperature. In general comment upon the relationship between the variance of the transformed variable and the original as it relates to the slope parameter.