Stats

Chapter 3

Descriptive Measures

• Measures of Central Tendency (measures of center)

– Mean– Median– ModeEx 1) 2, 3, 3, 7, 8, 9, 13, 13, 19

Ex 2) 4, 4, 6, 8, 11, 13, 14, 18

Resistant Measure

• http://www.shodor.org/interactivate/activities/PlopIt/

• Resistant measures• Trimmed mean—remove a percentage of

the smallest and largest values before computing the mean.

Notation• Σ greek letter sigma…”summation notation” It

means add up everything….

Ex: Let x represent the number of house plants a group of people have: 0, 0, 4, 3, 0, 2, 1, 7. We use xi to represent the “ith” observation….so x1 = 0, x2 = 0, x3 = 4…

So Σ xi means to add all the observations relating to x (number of house plants)

Sample Mean

• X =

Ex: Monthly tornado touchdowns in the U.S.3, 2, 47, 118, 204, 97, 68, 86, 62, 57, 98, 99

Find the mean, median, and mode(s) of the monthly tornado touchdowns in the U.S.

nxi

• You are in the market for buying a house. You are not a snob, but you would like to live in a “nice” neighborhood. You are looking at some different properties with your realtor today and they tell you that the average income for the neighborhood that you are looking at is $150,000 a year. That clinches your interest in living there. You buy the home and that nice figure sticks with you and gives you peace of mind.

• A year later, you are working with your realtor as part of a taxpayers’ committee. Your realtor is talking about the need to keep the tax rate down because the members of the neighborhood cannot afford it: After all, the average income in this neighborhood is only $35,000 a year. You can’t believe your ears….your realtor was not lying when you bought your house nor is he lying now.

 • How can this be possible (the members of the community have not

changed since you moved in)? Give an example that demonstrates how this could happen.

 • What implications can be made about the word average?

Measures of Variation

• Basketball Players Example• 72, 73, 76, 76, 78 and 67, 72, 76, 76, 84

• Range—The difference of between the maximum and minimum

• Sample Standard Deviation (takes into account all the data points) How far, on average, the observations are from the mean.

Sample Standard Deviation

1)( 2

nxxs i

x (x – x) (x – x)2

http://easycalculation.com/statistics/statistics.php

Standard Deviation

• Sample variance (the standard deviation without the square root)

• So what?• Three-Standard-Deviations Rule:

– Almost all observations in a data set lie within 3 standard deviations on either side of the mean

Excel Work/TI calculator work

• Excel Data Set 3.101– Mean– Median– Mode– Range– Standard Deviation

• TI calculators use following data:

3, 3.2, 4, 7.1, 3.3, 8, 1– Mean– Median– Mode– Range– Standard Deviation

Dividing Data Sets

• Percentiles (divided into hundredths)• Deciles (divided into tenths)• Quartiles (divided into 4ths)

– First quartile: 25/75 (the median of the part of the data set that lies at or below the median of the entire data set)

– Second quartile: (median) 50/50– Third quartile: 75/25 (the median of the part of the data

set that lies at or above the median of the entire data set)

Quartiles

Interquartile Range (IQR)– The difference between the first and third

quartiles: Q3 – Q1•The five-number summary is:

– Min, Q1, Q2, Q3, max

Five Number Summary

• Outlier: an observation that falls well outside the overall pattern of the data

• Use IQR to identify potential outliersLower Limit = Q1 – (1.5)(IQR)Upper Limit = Q3 + (1.5)(IQR)

• An observation that lies below the lower limit or above the upper limit MIGHT be an outlier.

Boxplots

• Boxplot (box and whisker): Displays the five-number summary – If there is a potential outlier (found by using

the IQR) the next adjacent value would be come the min or max and the outlier would be represented by *

Comparing Boxplots

• What can they show us? Car MPG grouped by country

Population vs Sample

Size mean

Sample n x

Population N μ

Size Standard Deviation

Sample n s

Population N σ

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2)(

Nxi

Parameter: descriptive measure for a population

Statistic: a descriptive measure for a sample

Standardized Variables

• Who has a higher score: a person who got a 26 on their ACT or a person who got a 632 on their SAT?

– ACT μ= 21.0, σ = 5.2– SAT μ= 518, σ = 115

Standardized Variable

xz

A z-score (standardized score) allows us to COMPARE…it puts everything into terms of the number of standard deviations a value is away from the mean.

Interpret z-scores

ACT μ= 21.0, σ = 5.2

• What actual ACT score would a z-score of 1.3 be?

• A junior’s standardized ACT score was -2.4. Should this student be concerned about getting into college? Why or why not?