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**Describing Distributions With Numbers**

Section 1.3 cont. (five number summary, boxplots, variance, standard deviation) Target Goal: I can calculate a 5 number summary and construct a boxplot. I can describe spread using the standard deviation of a distribution. Hw: pg 71: 92, 93, 95, 96, 97, 103, 105,

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Five-Number Summary Data set consisting of smallest observation, first quartile, median, third quartile, and largest observation written in order. Min Q1 M Q3 Max It gives us a quick summary of both center and spread.

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Bonds: Min Q1 M Q3 Max

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**Box (and whiskers) Plot**

A graph of a five-number summary of a distribution; best for side- by-side comparisons since they show less detail than histograms or stemplots; drawn either horizontally or vertically.

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Modified Boxplot Because the regular boxplot conceals outliers we will use modified boxplot. Plots outliers as isolated points Extend “whiskers” out to largest and/or smallest data points that are not outliers Remember: label axis, title graph, scale axis.

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**Regular (a) and modified (b) boxplots comparing Barry Bonds and Hank Aaron home runs.**

Min Q1 M Q3 Max Outlier

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**Activity: Acing the First Test Enter the scores of Mrs**

Activity: Acing the First Test Enter the scores of Mrs. Liao’s students on their first statistics test into L1 from page 71, ex. 92 Sort Data(ascending): Inspire Place cursor on column title Select:Menu,1:Actions,6:sort, sort by (a) Inspire: Appendix A6

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**Calculator: 1 VAR STAT(L1) 43, 82, 87.75, 93, 98 **

a. Find the five-number summary and verify your expectation from a. Calculator activity Enter the scores into L1 from page 71. Calculator: 1 VAR STAT(L1) 43, 82, 87.75, 93, 98 mean = 2544/30 (or )= 84.8 the median is greater than the mean

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**Between Q1 and Q3: Between 82 and 93**

b. What is the range of the middle half of the score of the statistic students? Between Q1 and Q3: Between 82 and 93

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**Acing the First Test Cont.**

Construct by hand a modified boxplot of the stats students scores. First find potential outliers. IQR = Q1 - IQR x 1.5 = Q3 + IQR x 1.5 = Outliers: Graph: Mark a small x for the outlier(s), next lowest min, Q1, M, Q3, max. Draw box and whisker plot.

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**Acing the First Test Cont.**

On your calculator: First define Plot1 to be a modified boxplot using the list. Graph, trace and compare. Is there an outlier? If so, was it the same as in part a ? Based on the boxplot, conjecture the shape of the corresponding histogram. Histogram shape:______________________

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**Acing the First Test Cont.**

Next, Define Plot2 to be a histogram also using the same list. Trace and compare. Did you guess correctly? Roughly draw histogram below.

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Important Note: If a distribution contains outliers, use the median and the IQR to describe the distribution.

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**The most common numerical description of a distribution is the :**

Standard deviation (s): measures spread by looking at how far the observations are from their mean The standard deviations (s) is the square root of the variance (s2).

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Variance (s2) of a set of observations is the average of the squares of the deviations of the observations from their mean. Note: Most of the time we will use calculator (STAT:CALC:1VAR STAT).

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**Why square the deviations?**

It makes them all non negative so that the observations far from the mean in either direction will have large positive squared deviation.

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**Properties of the Standard Deviation**

The sum of the deviations of the observations from their mean will always be zero. Choose s only when mean is chosen as the measure of center. s = 0 only when there is no spread (all observations have the same value). s, like the mean is not resistant. Strong skewness or a few outliers can make s very large. If a value is more than 2σ’s from the mean it is an outlier.

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**Why divide by (n – 1)? Degrees of freedom –**

Since is the exact balancing point of the data, the data will almost always be closer to , on average, than they will be to μ. The sum of the squared deviations of will underestimate the sum of the squared deviations of µ. To correct this we divide by n-1 instead of n.

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Example: Roger Maris New York Yankee Roger Maris held the single-season home run record from 1961 until Here are Maris’s home run counts for his 10 years in the American League:

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Maris’s mean number of home runs is = Find the standard deviation s from its definition (by hand). ∑ (xi - )2 = ( )2 + ( )2… s2 = / n-1 s2 = /9 s2 = s =

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b. Use your calculator to verify your results. (STAT:CALC:1 var stat:L1) Then use your calculator to find the mean and s for the 9 observations that remain when you leave out any outlier(s). Recall IQR x 1.5 Note: they choose 61 as an outlier while the upper bound is 61.5.

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Mean = 22.2 Sx = How does the leaving out the “outlier” affect the values of the mean and s? It caused the values of both measures to decrease. Is s a resistant measure of spread? Clearly, s is not a resistant measure of spread.

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Key Points of Chapter

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