# Chapter 3 Descriptive Statistics: Numerical Measures. Learning objectives

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1 Chapter 3 Descriptive Statistics: Numerical Measures Slide 1 Learning objectives 1. Single variable Part I (Basic) 1.1. How to calculate and use the measures of location 1.. How to calculate and use the measures of variability. Single variable Part II (Application).1. Understand what the measures of location (e.g., mean, median, mode) tell us about distribution shape -Discuss its use in manipulating simulated experiments.. How to detect outliers using z-score and empirical rule.3. How to use Box plot to explore data.4. How to calculate weighted mean.5. How to calculate mean and variance for grouped data 3. Two variables 3.1. How to calculate and use the measures of association -Covariance, Correlation coefficient Slide 1

2 L.O. 1. Numerical measures Part I Numerical measures Measures of Location Mean, median, mode, percentiles, quartiles Measures of Variability Range, interquartilerange, variance, standard deviation, coefficient of variation Slide 3 Numerical Measures If the measures are computed for data from a sample, they are called sample statistics. If the measures are computed for data from a population, they are called population parameters. A sample statistic is referred to as the point estimator of the corresponding population parameter. Slide 4

3 Mean The mean of a data set is the average of all the data values. The sample mean x is the point estimator of the population mean µ. L.O Mean Median Mode Percentile Quartile x x i n Sum of the values of the n observations Number of observations in the sample µ x i N Sum of the values of the N observations Number of observations in the population Slide 5 Median The median of a data set is the value in the middlewhen the data items are arranged in ascending order. For odd number of observations: the median is the middle value For even number of observations: the median is the average of the middle two values. L.O Mean Median Mode Percentile Quartile Whenever a data set has extreme values, the median is the preferred measure of central location. Often used in annual income and property value data Slide 6 3

4 Mode L.O Mean Median Mode Percentile Quartile The mode of a data set is the value that occurs with the greatest frequency. The greatest frequency can occur at two or more different values. If the data have exactly two modes, the data are bimodal. If the data have more than two modes, the data are multimodal. Slide 7 Example L.O Mean Median Mode Percentile Quartile Q4 (p. 84) Compute the mean, median, and mode of the following sample: 53, 55, 70, 58, 64, 57, 53, 69, 57, 68, 53 Mean Median 57 Mode 53 What is the median, if 59 is added to the data? Median 57.5 (57+58)/ Slide 8 4

5 Percentiles L.O Mean Median Mode Percentile Quartile A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. Admission test scores for colleges and universitiesare frequently reported in terms of percentiles. The pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more. Slide 9 Percentiles L.O Mean Median Mode Percentile Quartile Arrange the data in ascending order. Compute index i, the position of the pth percentile. i (p/100)n If i is not an integer, round up. The pth percentile is the value in the ith position. If i is an integer, the pth percentile is the average of the values in positions i and i+1. Slide 10 5

6 Quartiles L.O Mean Median Mode Percentile Quartile Quartiles are specific percentiles. First Quartile 5th Percentile Second Quartile 50th Percentile Median Third Quartile 75th Percentile Slide 11 Example: Percentiles and Quartiles L.O Mean Median Mode Percentile Quartile Q4 (p. 84) Find 5 th and 75 th percentiles from the sample below: 53, 55, 70, 58, 64, 57, 53, 69, 57, 68, 53 5 th percentile First quartile th percentile Third quartile 68 Slide 1 6

7 Measures of Variability It is often desirable to consider measures of variability(dispersion), as well as measures of location. For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each. Range Interquartile Range Variance Standard Deviation Coefficient of Variation Slide 13 Range The range of a data set is the difference between the largest and smallest data values. It is the simplest measure of variability. It is very sensitive to the smallest and largest data values. L.O. 1.. Range IQR Variance St. Deviation Coefficient of variation Range of the sample: 53, 55, 70, 58, 64, 57, 53, 69, 57, 68, Slide 14 7

8 Interquartile Range (IQR) The interquartilerange of a data set is the differencebetween the third quartile and the first quartile. It is the range for the middle 50% of the data. It overcomes the sensitivity to extreme data values. L.O. 1.. Range IQR Variance St. Deviation Coefficient of variation IQR of the sample: 53, 55, 70, 58, 64, 57, 53, 69, 57, 68, Slide 15 Variance The variance is a measure of variability that utilizes all the data. L.O. 1.. Range IQR Variance St. Deviation Coefficient of variation The variance is the average of the squared differences between each data value and the mean. The variance is computed as follows: s ( x i x ) n 1 for a sample σ ( x µ ) i N for a population Slide 16 8

9 Standard Deviation L.O. 1.. Range IQR Variance St. Deviation Coefficient of variation The standard deviation of a data set is the positive square root of the variance. It is measured in the same units as the data, making it more easily interpreted than the variance. The standard deviation is computed as follows: s s σ σ for a sample for a population Slide 17 Coefficient of Variation L.O. 1.. Range IQR Variance St. Deviation Coefficient of variation The coefficient of variation indicates how large the standard deviation is in relation to the mean. The coefficient of variation is computed as follows: s 100 x % for a sample σ 100 µ % for a population Slide 18 9

10 Example: Variance, Standard Deviation, And Coefficient of Variation Consider the same data set: 53, 55, 70, 58, 64, 57, 53, 69, 57, 68, 53 Variance ( x i x ) s n L.O. 1.. Range IQR Variance St. Deviation Coefficient of variation Standard Deviation s s Coefficient of Variation the standard deviation is about 11% of of the mean s % 100 % 11.8% x Slide 19 L.O.. Numerical measure Part II Measures of Distribution Shape Detecting Outliers z-score, empirical rule Exploratory Data Analysis The Weighted Mean and Working with Grouped Data Slide 0 10

11 Distribution Shape Symmetric (not skewed) Skewness is zero. Mean and median are equal. Relative Frequency Slide 1 Distribution Shape Moderately Skewed Left Skewness is negative. Mean will usually be less than the median. Relative Frequency Slide 11

12 Distribution Shape Moderately Skewed Right Skewness is positive. Mean will usually be more than the median. Relative Frequency Slide 3 Distribution Shape Highly Skewed Right Skewness is positive (often above 1.0). Mean will usually be more than the median. Relative Frequency Slide 4 1

13 z-scores The z-score is often called the standardized value. It denotes the number of standard deviations a data value x i is from the mean. z i x i x s Slide 5 z-scores An observation s z-score is a measure of the relative location of the observation in a data set. A data value less than the sample mean will have a z-score less than zero. A data value greater than the sample mean will have a z-score greater than zero. A data value equal to the sample mean will have a z-score of zero. Slide 6 13

14 Empirical Rule For data having a bell-shaped distribution: 68.6% of the values of a normal random variable are within +/- 1 standard deviation ofits mean % of the values of a normal random variable are within +/- standard deviations ofits mean. 99.7% of the values of a normal random variable are within +/- 3 standard deviations ofits mean. Slide 7 Empirical Rule 99.7% 95.44% 68.6% µ 3σ µ 1σ µ σ µ µ + 1σ µ + 3σ µ + σ x Slide 8 14

15 Detecting Outliers An outlier is an unusually small or unusually large value in a data set. A data value with a z-score less than -3 or greater than +3 might be considered an outlier. It might be: an incorrectly recorded data value a data value that was incorrectly included in the data set a correctly recorded data value that belongs in the data set Slide 9 Exploratory Data Analysis The techniques of exploratory data analysis consist of simple arithmetic and easy-to-draw pictures that can be used to summarize data quickly. Five-Number Summary Box Plot Slide 30 15

16 Five-Number Summary Sample: 53, 55, 70, 58, 64, 57, 53, 69, 57, 68, 53 1 Smallest Value First Quartile Median Third Quartile Largest Value Slide 31 Box Plot A box plot is based on a five-number summary. Outlier Lower limit 30.5 Whisker Upper limit *IQR 1.5*IQR * No whisker this side : smallest value Q1 Q1 53 Q3 68 Q 57 Largest value (70) Slide 3 16

17 Weighted Mean The Weighted Mean and Working with Grouped Data Mean for Grouped Data Variance for Grouped Data Standard Deviation for Grouped Data Slide 33 Weighted Mean When the mean is computed by giving each data value a weight that reflects its importance, it is referred to as a weighted mean. Class grade is usually computed by weighted mean. In class midterm exam Descriptive statistics and distributions 40% Final group project Statistical inference 30% Group project presentation 10% Homework 10% Participation 10% weight When data values vary in importance, the analyst must choose the weight that best reflects the importance of each value. Slide 34 17

18 Weighted Mean x wx w i i wi where: x i value of observation i w i weight for observation i Slide 35 Grouped Data The weighted mean computation can be used to obtain approximations of the mean, variance, and standard deviation for the grouped data. To compute the weighted mean, we treat the midpoint of each class as though it were the mean of all items in the class. We compute a weighted mean of the class midpoints using the class frequencies as weights. Similarly, in computing the variance and standard deviation, the class frequencies are used as weights. Slide 36 18

19 Mean for Grouped Data Sample Data fm x n i i Population Data where: µ f M i N f i frequency of class i M i midpoint of class i i Slide 37 Sample Mean for Grouped Data Given below is the previous sample of monthly rents for 70 efficiency apartments, presented here as grouped data in the form of a frequency distribution. Rent (\$) Frequency Slide 38 19

20 Sample Mean for Grouped Data Rent (\$) f i Total 70 M i f i M i ,55 x This approximation differs by \$.41 from the actual sample mean of \$ Slide 39 For sample data Variance for Grouped Data s For population data f i ( M i x ) n 1 σ f i ( M i µ ) N Slide 40 0

21 Sample Variance for Grouped Data Rent (\$) f i Total 70 M i M i - x (M i - x) f i (M i - x) continued Slide 41 Sample Variance for Grouped Data Sample Variance s 08,34.9/(70 1) 3, Sample Standard Deviation s 3, This approximation differs by only \$.0 from the actual standard deviation of \$ Slide 4 1

22 Covariance L.O. 3. Measures of Association Between Two Variables Correlation Coefficient Slide 43 Covariance L.O. 3. Covariance Correlation The covariance is a measure of the linear association between two variables. Positive values indicate a positive relationship. Negative values indicate a negative relationship. Slide 44

23 Covariance L.O. 3. Covariance Correlation The correlation coefficient is computed as follows: s xy ( x i x )( y i y ) n 1 for samples σ xy ( x i µ x )( y i µ y ) N for populations Slide 45 Correlation Coefficient L.O. 3. Covariance Correlation The coefficient can take on values between -1 and +1. Values near -1 indicate a strong negative linear relationship. Values near +1 indicate a strong positive linear relationship. Slide 46 3

24 Correlation Coefficient L.O. 3. Covariance Correlation The correlation coefficient is computed as follows: r xy s xy ρ ss x y xy σ xy σσ x y for samples for populations Slide 47 Correlation Coefficient L.O. 3. Covariance Correlation Correlation is a measure of linear association and not necessarily causation. Just because two variables are highly correlated, it does not mean that one variable is the cause of the other. Slide 48 4

25 In class Exercise L.O. 3. Covariance Correlation Q45 (p. 11) Q46 (p. 11) Slide 49 End of Chapter 3 Slide 50 5

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