Center and Spread. Measures of Center and Spread. Example: Mean. Mean: the balance point 2/22/2009. Describing Distributions with Numbers.

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1 Chapter 3 Section3-: Measures of Center Section 3-3: Measurers of Variation Section 3-4: Measures of Relative Standing Section 3-5: Exploratory Data Analysis Describing Distributions with Numbers The overall pattern of the distribution of a quantitative variable is described by its shape, center, and spread. By inspecting the histogram we can describe the shape of the distribution, but as we saw, we can only get a rough estimate for the center and spread. We need a more precise numerical description of the center and spread of the distribution. 1 Center and Spread Measures of Center and Spread Wide spread Narrow spread Mean Median Mode CENTER SPREAD Range Inter-quartile range (IQR) Variance/Standard deviation About the same center Mean: the balance point Mean: x x-bar the sum of the observations divided by the number of observations). If the n observations are x 1, x, x 3,K, x n then n xi x1 + x + x3 + K+ xn i x = = = 1 n n Example: Mean x = 5, x = 7, x = 3, x = 38, x = 7 1 3, 4 5 The meanis x x1 + x + x3 + x4 + x5 i= 1 x = = n n 5 x = = 1 hours 5 5 i 1

2 Median The median M is the midpoint of the distribution (like the median strip in a road) It is the number such that half of the observations fall above and half fall below. How to find the median? 1.Order the data from smallest to largest.. If n is odd, the median M is the center observation in the ordered list. This observation is the one "sitting" in the (n+1)/ spot in the ordered list.. If n is even, the median M is the mean of the two center observations in the ordered list. These two observations are the ones "sitting" in the n/ and n/ + 1 spots in the ordered list. Median Example: Median x = 5, x = 7, x = 3, x = 38, x = 7 1 3, 4 5 Step 1:order the data: Median = 7 Example : Median If the data are x = 5, x = 7, x = 3, x = , 4 Step 1:order the data: Data set A: median is 68 mean is 68.1 Example outlier Data set B: median is still 68 mean is 16 Median = = 6 The mean is very sensitive to outliers, while the median is resistant to outliers.

3 Comparing the mean and the median Mean describes the center as an average value, where the actual values of the data points play an important role Sensitive to outliers Median locates the middle value as the center, and the order of the data is the key to finding it. Not sensitive to outliers Symmetric distributions with no outliers Mean Median Left-skewed distributions Right-skewed distributions The tail pulls the mean. The tail pulls the mean. Mean < Median Median=68 Mean=7 Median< Mean Which measure of center to use? We will therefore use the meanas a measure of center for symmetric distributions with no outliers. Otherwise, the median will be a more appropriate measure of the center of our data. Another measure of center: Mode Mode is the most frequent value in the data set Data:, 4, 4, 4, 5, 5, 6, 7, 8, 10, 15 Mode = 4 Data:, 4, 4, 4, 5, 5, 5, 6, 7, 8, 10, 15 Mode = 4, 5 Data:, 4, 5, 6, 7, 8, 10, 15 No mode 3

4 Summary: Measures of Center The three main numerical measures for the center of a distribution are the mean x, the medianm, and the mode. The mean is the average value, while the median is the middle value. The mode is the most frequent value. The mean is very sensitive to outliers, while the median is resistant to outliers. The mean is an appropriate measure of center only for symmetric distributions with no outliers. In all other cases, the median should be used to describe the center of the distribution. Spread Spread: how far from the center the data tend range. If all the data points are identical, there would be no spread at all. Numerically, the spread would be zero. Ex.: Center: 5 Spread: 0 Measures of Spread Measures of Spread Range, Inter-quartile range (IQR), Variance / Standard deviation These measures provide different ways to quantify the variability of the distribution. Range Range = max. value min. value Inter-quartile range (IQR) the IQR gives the range covered by the MIDDLE 50% of the ordered data Example: Data:, 4, 4, 4, 5, 5, 6, 7, 8, 10, 15 Range= max.-min. = 15- = 13 4

5 How to find the IQR? Step 1:arrange the data in increasing order Step :find the median How to find the IQR? Step 3: Find the median of the lower 50% of the data. This is called the first quartile of the distribution and is denoted by Q1. How to find the IQR? Step 4: Repeat this again for the top 50% of the data. Find the median of the top 50% of the data. This is called the third quartile of the distribution and is denoted by Q3. IQR The middle 50% of the data falls between Q1 and Q3, and therefore: IQR = Q3 -Q1 IQR Example Weights of 10 students: 10, 118, 10, 136, 138, 149, 157, 157, 161, M = = IQR = = 37 5

6 Note The IQR should be used as a measure of spread of a distribution only when the median is used as a measure of center. Median IQR Using the IQR to detect outliers The 1.5(IQR) Criterion for Outliers An observation is considered a suspected outlier if it is below Q1-1.5(IQR) or above Q (IQR) The 1.5(IQR) Criterion for Outliers Example 1 Weights of 10 students: 10, 118, 10, 136, 138, 149, 157, 157, 161, 15 + M = = IQR = = 37 Q IQR = ( 37) = 1. 5 Anything above 1.5? YES. 15 IS an outlier. Example Outliers! Outlier! Data: -15, 8, 9, 1, 14, 19,, 3, 3, 45, 50 M IQR= 3-9 = (IQR)= 1.5 (14) =1 Anything below Q1-1.5(IQR)=9 1 = -1? Anything above Q3+1.5(IQR)=3 +1 = 44? YES! YES! Five-number summary To get a quick summary of both center and spread, we consider these five numbers: Mininum value Q1 Median Five-number summary Q3 Maximum value 6

7 Boxplot John Tukeyinvented another kind of display to show off the fivenumber summary. It s called boxplot. Example 1 Weights of 10 students: 10, 118, 10, 136, 138, 149, 157, 157, 161, 180 Min. + M = = Max Example Outlier Weights of 10 students: 10, 118, 10, 136, 138, 149, 157, 157, 161, 15 Min. + M = = New Max * Example Largest = max = 6.1 = third quartile = 4.35 M= median = 3.4 = first quartile =. Smallest = min = 0.6 Years until death Disease X Five-number summary Interpretation Comparing distributions Low variability High variability Boxplotsare best used for side-byside comparison of more than one distribution. 7

8 Summary Measures of the center of distributions: Mean Median Mode Measures of spread of distributions: Range IQR Using IQR to detect outliers the 1.5(IQR) rule Boxplots Variance/Standard deviation Variance and Standard Deviation: The idea The standard deviation gives the average (or typical distance) between a data point and the mean, x The formulas We have n observations: 1 Variance: ( x1 x) + ( x x) + K+ ( xn x) s = n 1 Standard deviation: s = x, x, K, x n ( x1 x) + ( x x) + K+ ( xn x) n 1 Example: Video Store Customers The following are the number of customers who entered a video store in 8 consecutive hours: 7, 9, 5, 13, 3, 11, 15, 9 Hour 1 st nd 3 rd 4 th 5 th 6 th 7 th 8 th # of customers x = 7, x = 9, x = 5, x = 13, x = 3, x = 11, x = 15, x = Find the mean and the standard deviation of the distribution Dotplot and the Mean 7 x = = 9 8 Mean = 9 Standard deviation: Steps 1,, and 3 Observations x i Deviations xi x Squared deviations ( xi x) = - (-) = = 0 (0) = = -4 (-4) = = 4 (4) = = -6 (-6) = = () = = 6 (6) = = 0 (0) = 0 Mean: 9 Sum= 0!!!!!!!!!! Sum=

9 Variance s Standard deviation s Steps 4, and 5 ( x x) + ( x x) + K+ ( xn x) = n s = 8 1 = 16 s = = Variance = 16 = 4 Variance The typical distance from the mean is 4. FAQ about the Standard Deviation 1. Why do we need to square the deviations? Because the sum of the deviations from the mean is ALWAYS 0!. Why do we divide by n-1 and not by n? Because we know (question 1) that the sum of the deviations is always 0, so that knowing n-1 of them determines the last one. Only n-1 of the squared deviations can vary freely. The number n-1 is called the degrees of freedom FAQ about the Standard Deviation 3. Why do we take the square root? s =16 is an average of the squared deviations, and therefore has different units of measurement. In this case 16 is measured in "squared customers", which obviously cannot be interpreted. We therefore, take the square root in order to go back to the original units of measurement. Facts about the standard deviation (s) smeasures the spread about the mean and should be used only when the mean is chosen as the measure of center. That is, when the distribution of the data is roughly symmetric with no outliers. Mean Standard deviation 51 5 Facts about the Standard Deviation (s) sis always zero or greater than 0. s= 0 only when there is no spread, i.e., the data values are identical. sgets larger as the spread increases. shas the same units of measurements as the original observations. Like the mean, sis not resistant. It is very sensitive to outliers. Calculator (TI-83, TI-84) Steps: 1. Enter your data into a List: STAT EDIT 1: Edit Enter you data into L1. Find the mean, median, standard deviation, five-number summary STAT CALC 1: 1-Var Stats You see in your window 1-Var Stats (L1) nd

10 Try it: 3, 18, 19, 5, 7, 7, 0, 17, 4 x = Mean x = 190 x Sx = σx = n = 9 min X = 7 Q = 185. Med = 3 Q 1 3 = 43 = 6 max X = 7 Standard deviation Number of entries Five-number summary Measures of Relative Standing We can compare values from different data sets using z-scores: x mean z = s. d. A z-score measures the number of standard deviations that a data value x is from the mean. Ordinary values: - z-score Unusual values: z-score < - or z-score > 55 Example IQ scores have a mean of 100 and a standard deviation of 16. Albert Einstein reportedly had an IQ of 160. Is Einstein s IQ score unusual? x mean z = = s. d = Since the z-score is higher than, we can conclude that Einstein s IQ score is unusual. Median Find the median of the following 9 numbers: a)65 b)64 c)67 d) Median For the data in the previous question, Suppose that the last data point is actually 115 instead of 85. What effect would this new maximum have on our value for the median of the dataset? a)increase the value of the median. b)decrease the value of the median. c) Not change the value of the median. Mean For the data in the previous question, Suppose that the last data point is actually 115 instead of 85. What effect would this new maximum have on our value for the mean of the dataset? a) Increase the value of the mean. b) Decrease the value of the mean. c) Not change the value of the mean

11 Mean vs. median For the dataset volumes of milk dispensed into -gallon milk cartons, should you use the mean or the median to describe the center? a) Mean b) Median Mean vs. median For the dataset sales prices of homes in Los Angeles, should you use the mean or the median to describe the center? a) Mean b) Median 61 6 Mean vs. median For the dataset incomes for people in the United States, should you use the mean or the median to describe the center? a) Mean b) Median Boxplots You have a boxplotfor the tar content of 5 different cigarettes. What is a plausible set of values for the five-number summary? a) Min = 13, Q1 = 10, Median = 1.6, Q3 = 14, Max = 15 b) Min = 1, Q1 = 8.5, Median = 1.6, Q3 = 15, Max = 17 c) Min = 1, Q1 = 8.5, Median = 11.5, Q3 = 13, Max = 15 d) Min = 8.5, Q1 = 10, Median = 11.5, Q3 = 15, Max = Boxplots The shape of the boxplotbelow can be described as: a) Bi-modal b) Left-skewed c) Right-skewed d) Symmetric e) Uniform Side-by-side boxplots Look at the following side-by-side boxplots and compare the female and male shoulder girth. a) Females have a typically smaller shoulder girth than males. b) Females have a typically larger shoulder girth than males. c) Females and males have about the same shoulder girths

12 Side-by-side boxplots Look at the following side-by-side boxplotsand compare the female and male thigh girth. Comparing two histograms Compare the centers of Distr. A (Female Shoulder Girth) and Distr. B (Male Shoulder Girth) shown below. a) Females have a typically smaller thigh girth than males. b) Females have a typically larger thigh girth than males. c) Females and males have about the same thigh girth. 67 a) The center of Distr. A is greater than the center of Distr. B. b) The center of Distr. A is less than the center of Distr. B. c) The center of Distr. A is equal to the center of Distr. B. 68 Comparing two histograms Compare the spreads of Distr.A(Female Shoulder Girth) and Distr. B (Male Shoulder Girth) shown below. Boxplots What is the approximate range of the Male Wrist Girth dataset shown below? a) The spread of Distr. A is greater than the spread of Distr. B. b) The spread of Distr. A is less than the spread of Distr. B. c) The spread of Distr. A is equal to the spread of Distr. B. a) 14.5 to 19.5 b)16.5 to 17 c) 16.5 to 18 d)17 to 19.5 e) 14.5 to 16.5 and 18 to Boxplots What is the approximate interquartilerange of the Male Wrist Girth dataset shown below? Outliers If a dataset contains outliers, which measure of spread is resistant? a) 14.5 to 19.5 b) 16.5 to 17 c) 16.5 to 18 d) 17 to 19.5 e) 14.5 to 16.5 and 18 to 19.5 a) Range b) Interquartile range c) Standard deviation d) Variance

13 Standard deviation Which of the following statements is TRUE? a) Standard deviation has no unit of measurement. b) Standard deviation is either positive or negative. c) Standard deviation is inflated by outliers. d)standard deviation is used even when the mean is not an appropriate measure of center. Center and spread For the following distribution of major league baseball players salaries in 199, which measures of center and spread are more appropriate? a)mean and standard deviation b)median and interquartile range