# Chapter 3: Numerical Descriptive Measures Chapter 3 Numerical Descriptive Measures Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 1 Objectives In this chapter, you learn to: Describe the properties of central tendency, variation, and shape in numerical data Construct and interpret a boxplot Compute descriptive summary measures for a population Calculate the covariance and the coefficient of correlation Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 2 Summary Definitions DCOVA The central tendency is the extent to which the values of a numerical variable group around a typical or

central value. The variation is the amount of dispersion or scattering away from a central value that the values of a numerical variable show. The shape is the pattern of the distribution of values from the lowest value to the highest value. Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 3 Measures of Central Tendency: The Mean DCOVA The arithmetic mean (often just called the mean) is the most common measure of central tendency For a sample of size n: Pronounced x-bar

The ith value n X X i1 n i X1 X 2 Xn n Sample size Copyright 2016 Pearson Education, Ltd. Observed values Chapter 3, Slide 4 Measures of Central Tendency: The Mean (cont) DCOVA

The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) 11 12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 17 18 19 20 Mean = 13 11 12 13 14 15 65 13 5 5 Mean = 14 11 12 13 14 20 70 14 5 5 Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 5 Numerical Descriptive Measures for a Population DCOVA

Descriptive statistics discussed previously described a sample, not the population. Summary measures describing a population, called parameters, are denoted with Greek letters. Important population parameters are the population mean, variance, and standard deviation. Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 6 Numerical Descriptive Measures for a Population: The mean DCOVA The population mean is the sum of the values in the population divided by the population size, N N X Where

i1 N i X1 X 2 XN N = population mean N = population size Xi = ith value of the variable X Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 7 Measures of Central Tendency: The Median DCOVA In an ordered array, the median is the middle number (50% above, 50% below) 11 12 13 14 15 16 17 18 19 20 11 12 13 14 15 16 17 18 19 20

Median = 13 Median = 13 Less sensitive than the mean to extreme values Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 8 Measures of Central Tendency: Locating the Median DCOVA The location of the median when the values are in numerical order (smallest to largest): n 1 position the ordered data If theMedian number of values is odd, the median position is the middlein number

2 If the number of values is even, the median is the average of the two middle numbers Note that is not the value of the median, only the position of the median in the ranked data n 1 2 Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 9 Measures of Central Tendency: The Mode DCOVA Value that occurs most often Not affected by extreme values Used for either numerical or categorical data

There may be no mode There may be several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 Copyright 2016 Pearson Education, Ltd. 0 1 2 3 4 5 6 No Mode Chapter 3, Slide 10 Measures of Central Tendency: Review Example DCOVA House Prices: \$2,000,000 \$ 500,000 \$ 300,000 \$ 100,000 \$ 100,000 Sum \$ 3,000,000 Mean: (\$3,000,000/5) = \$600,000 Median: middle value of ranked data = \$300,000 Mode: most frequent value

= \$100,000 Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 11 Measures of Central Tendency: Which Measure to Choose? DCOVA The mean is generally used, unless extreme values (outliers) exist. The median is often used, since the median is not sensitive to extreme values. For example, median home prices may be reported for a region; it is less sensitive to outliers. In some situations it makes sense to report both the mean and the median. Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 12

Measures of Central Tendency: Summary DCOVA Central Tendency Arithmetic Mean Median Mode n X X i1 n i Middle value in the ordered array Copyright 2016 Pearson Education, Ltd. Most frequently

observed value Chapter 3, Slide 13 Shape of a Distribution DCOVA Describes how data are distributed Two useful shape related statistics are: Skewness Measures the extent to which data values are not symmetrical Kurtosis Kurtosis affects the peakedness of the curve of

the distributionthat is, how sharply the curve rises approaching the center of the distribution Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 14 Shape of a Distribution (Skewness) DCOVA Measures the extent to which data is not symmetrical Left-Skewed Symmetric Right-Skewed Mean < Median Mean = Median Median < Mean Skewness Statistic

<0 0 Copyright 2016 Pearson Education, Ltd. >0 Chapter 3, Slide 15 Measures of Variation DCOVA Variation Range Variance Standard Deviation Measures of variation give information on the spread or variability or dispersion of the data values. Copyright 2016 Pearson Education, Ltd. Coefficient of Variation

Same center, different variation Chapter 3, Slide 16 Measures of Variation: The Range DCOVA Simplest measure of variation Difference between the largest and the smallest values: Range = Xlargest Xsmallest Example: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 13 - 1 = 12 Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 17 Measures of Variation: Why The Range Can Be Misleading DCOVA

Does not account for how the data are distributed 7 8 9 10 11 12 7 8 Range = 12 - 7 = 5 9 10 11 12 Range = 12 - 7 = 5

Sensitive to outliers 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5 Range = 5 - 1 = 4 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120 Range = 120 - 1 = 119 Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 18 Measures of Variation: The Sample Variance DCOVA Average (approximately) of squared deviations of values from the mean n Sample variance: 2 (X S i1 Where

i X) 2 n -1 X = arithmetic mean n = sample size Xi = ith value of the variable X Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 19 Measures of Variation: The Sample Standard Deviation DCOVA Most commonly used measure of variation Shows variation about the mean Is the square root of the variance Has the same units as the original data n

Sample standard deviation: Copyright 2016 Pearson Education, Ltd. (X X) 2 i S i1 n -1 Chapter 3, Slide 20 Measures of Variation: The Standard Deviation DCOVA Steps for Computing Standard Deviation 1. Compute the difference between each value and the mean. 2. Square each difference. 3. Add the squared differences. 4. Divide this total by n-1 to get the sample variance. 5. Take the square root of the sample variance to get the sample standard deviation.

Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 21 Measures of Variation: Sample Standard Deviation: Calculation Example Sample Data (Xi) : DCOVA 10 12 14 n=8 15 17 18 18 24 Mean = X = 16

(10 X)2 (12 X)2 (14 X)2 (24 X)2 S n 1 (10 16)2 (12 16)2 (14 16)2 (24 16)2 8 1 130 7 4.3095 A measure of the average scatter around the mean Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 22 Measures of Variation: Comparing Standard Deviations DCOVA Data A 11 21 12 13

14 15 16 17 18 19 20 Mean = 15.5 S = 3.338 20 Mean = 15.5 S = 0.926 Data B 11 21 12 13

14 15 16 17 18 19 Data C 11 12 13 14 15 16 17 18

19 Copyright 2016 Pearson Education, Ltd. 20 21 Mean = 15.5 S = 4.567 Chapter 3, Slide 23 Measures of Variation: Comparing Standard Deviations DCOVA Smaller standard deviation Larger standard deviation Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 24 Numerical Descriptive Measures For A Population: The Variance 2 DCOVA Average of squared deviations of values from the mean N

Population variance: (X ) 2 i1 Where 2 i N = population mean N = population size Xi = ith value of the variable X Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 25 Numerical Descriptive Measures For A Population: The Standard Deviation DCOVA

Most commonly used measure of variation Shows variation about the mean Is the square root of the population variance Has the same units as the original data N Population standard deviation: Copyright 2016 Pearson Education, Ltd. 2 (X ) i i1 N Chapter 3, Slide 26 Sample statistics versus population parameters DCOVA

X 2 S2 S Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 27 Measures of Variation: Summary Characteristics DCOVA The more the data are spread out, the greater the range, variance, and standard deviation. The more the data are concentrated, the smaller the range, variance, and standard deviation.

If the values are all the same (no variation), all these measures will be zero. None of these measures are ever negative. Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 28 Measures of Variation: The Coefficient of Variation DCOVA Measures relative variation Always in percentage (%) Shows variation relative to mean Can be used to compare the variability of two or more sets of data measured in different units

S 100% CV X Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 29 Measures of Variation: Comparing Coefficients of Variation DCOVA Stock A: Average price last year = \$50 Standard deviation = \$5 S \$5 CVA 100% 100% 10% \$50 X Stock B: Average price last year = \$100 Standard deviation = \$5

S \$5 CVB 100% 100% 5% \$100 X Copyright 2016 Pearson Education, Ltd. Both stocks have the same standard deviation, but stock B is less variable relative to its price Chapter 3, Slide 30 Measures of Variation: Comparing Coefficients of Variation (cont) Stock A: Average price last year = \$50 Standard deviation = \$5 S \$5 CVA 100% 100% 10%

\$50 X Stock C: Average price last year = \$8 Standard deviation = \$2 S \$2 CVC 100% 100% 25% \$8 X Copyright 2016 Pearson Education, Ltd. DCOVA Stock C has a much smaller standard deviation but a much higher coefficient of variation Chapter 3, Slide 31 Quartile Measures

DCOVA Quartiles split the ranked data into 4 segments with an equal number of values per segment 25% 25% Q1 25% Q2 25% Q3 The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger Q2 is the same as the median (50% of the observations are smaller and 50% are larger) Only 25% of the observations are greater than the third quartile Copyright 2016 Pearson Education, Ltd.

Chapter 3, Slide 32 Quartile Measures: Locating Quartiles DCOVA Find a quartile by determining the value in the appropriate position in the ranked data, where First quartile position: Q1 = (n+1)/4 ranked value Second quartile position: Q2 = (n+1)/2 ranked value Third quartile position: Q3 = 3(n+1)/4 ranked value where n is the number of observed values Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 33 Quartile Measures: Calculation Rules

DCOVA When calculating the ranked position use the following rules If the result is a whole number then it is the ranked position to use If the result is a fractional half (e.g. 2.5, 7.5, 8.5, etc.) then average the two corresponding data values. If the result is not a whole number or a fractional half then round the result to the nearest integer to find the ranked position. Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 34 Quartile Measures: Locating Quartiles DCOVA Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 (n = 9)

Q1 is in the (9+1)/4 = 2.5 position of the ranked data so use the value half way between the 2nd and 3rd values, so Q1 = 12.5 Q1 and Q3 are measures of non-central location Q2 = median, is a measure of central tendency Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 35 Quartile Measures Calculating The Quartiles: Example DCOVA Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 (n = 9) Q1 is in the (9+1)/4 = 2.5 position of the ranked data, so Q1 = (12+13)/2 = 12.5 Q2 is in the (9+1)/2 = 5th position of the ranked data, so Q2 = median = 16

Q3 is in the 3(9+1)/4 = 7.5 position of the ranked data, so Q3 = (18+21)/2 = 19.5 Q1 and Q3 are measures of non-central location Q2 = median, is a measure of central tendency Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 36 Quartile Measures: The Interquartile Range (IQR) DCOVA The IQR is Q3 Q1 and measures the spread in the middle 50% of the data The IQR is also called the midspread because it covers the middle 50% of the data The IQR is a measure of variability that is not influenced by outliers or extreme values

Measures like Q1, Q3, and IQR that are not influenced by outliers are called resistant measures Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 37 Calculating The Interquartile Range DCOVA Example: X minimum Q1 25% 12 Median (Q2) 25% 30 25% 45

X Q3 maximum 25% 57 70 Interquartile range = 57 30 = 27 Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 38 The Five Number Summary DCOVA The five numbers that help describe the center, spread and shape of data are: Xsmallest First Quartile (Q1) Median (Q2) Third Quartile (Q3) Xlargest

Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 39 Relationships among the five-number summary and distribution shape DCOVA Left-Skewed Symmetric Right-Skewed Median Xsmallest Median Xsmallest Median Xsmallest > < Xlargest Median Xlargest Median Xlargest Median

Q1 Xsmallest Q1 Xsmallest Q1 Xsmallest > < Xlargest Q3 Xlargest Q3 Xlargest Q3 Median Q1 Median Q1 Median Q1 > <

Q3 Median Q3 Median Q3 Median Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 40 Five Number Summary and The Boxplot DCOVA The Boxplot: A Graphical display of the data based on the five-number summary: Xsmallest -- Q1 -- Median -- Q3 -- Xlargest Example: 25% of data Xsmallest Q1 25% of data

Median Copyright 2016 Pearson Education, Ltd. 25% of data Q3 25% of data Xlargest Chapter 3, Slide 41 Five Number Summary: Shape of Boxplots If data are symmetric around the median then the box and central line are centered between the endpoints Xsmallest DCOVA Q1 Median

Q3 Xlargest A Boxplot can be shown in either a vertical or horizontal orientation Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 42 Distribution Shape and The Boxplot Left-Skewed Q1 Q2 Q3 Symmetric Q1 Q2 Q3 Copyright 2016 Pearson Education, Ltd. DCOVA Right-Skewed

Q1 Q2 Q3 Chapter 3, Slide 43 Boxplot Example DCOVA Below is a Boxplot for the following data: Xsmallest 0 2 Q1 2 Q2 / Median 2 3 3 0 2 3 5

4 Q3 5 5 Xlargest 9 27 27 The data are right skewed, as the plot depicts Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 44 Locating Extreme Outliers: Z-Score (It is studied with Chapter 6) DCOVA

To compute the Z-score of a data value, subtract the mean and divide by the standard deviation. The Z-score is the number of standard deviations a data value is from the mean. A data value is considered an extreme outlier if its Z-score is less than -3.0 or greater than +3.0. The larger the absolute value of the Z-score, the farther the data value is from the mean. Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 45 Locating Extreme Outliers: Z-Score DCOVA X X Z S where X represents the data value X is the sample mean

S is the sample standard deviation Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 46 Locating Extreme Outliers: Z-Score DCOVA Suppose the mean math SAT score is 490, with a standard deviation of 100. Compute the Z-score for a test score of 620. X X 620 490 130 Z 1.3 S 100 100 A score of 620 is 1.3 standard deviations above the mean and would not be considered an outlier. Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 47

The Empirical Rule (It is studied with Chapter 6( DCOVA The empirical rule approximates the variation of data in a bell-shaped distribution Approximately 68% of the data in a bell shaped distribution is within 1 standard deviation of the mean or 1 68% 1 Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 48 The Empirical Rule DCOVA Approximately 95% of the data in a bell-shaped distribution lies within two standard deviations of the mean, or 2

Approximately 99.7% of the data in a bell-shaped distribution lies within three standard deviations of the mean, or 3 95% 99.7% 2 3 Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 49 Using the Empirical Rule DCOVA Suppose that the variable Math SAT scores is bellshaped with a mean of 500 and a standard deviation of 90. Then, Approximately 68% of all test takers scored between 410 and 590, (500 90).

Approximately 95% of all test takers scored between 320 and 680, (500 180). Approximately 99.7% of all test takers scored between 230 and 770, (500 270). Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 50 Chapter Summary In this chapter we have discussed: Describing the properties of central tendency, variation, and shape in numerical data Constructing and interpreting a boxplot Computing descriptive summary measures for a population Calculating the covariance and the coefficient of correlation Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 51