chapter 5 describing distributions numerically

Chapter 5Describing Distributions Numerically

AP Statistics

Center and Spread

• In this chapter, we will eliminate much of the vagueness of finding the center and spread of a distribution

• The values we use will be determined by the shape of the distribution

Skewed Distribution

• Center: Median• Spread: Interquartile Range (IQR)

• Both of these are “resistant”• Both should include units

Skewed Distribution

How to find the IQR 1. Find median 2. Find the median of both halves of data

the lower median is 1st Quartilethe upper median is 3rd Quartile

3. Subtract the two quartile scores** 1st Quartile = 25th percentile** 3rd Quartile = 75th percentile

Outliers

• Really just data that seems unusual• Formally we compute fences and if data point

is outside the fences, we consider it an outlier• Always use common sense

• Upper fence:• Lower fence: IQRQ

IQRQ5.15.1

1

3

5-Number Summary and Boxplots

The five number summary is used to create a boxplot

Boxplots

Boxplot

• Complement histograms by providing more specific information

• Look at histogram and boxplot together

• Most useful when comparing distributions

Symmetric Distributions

• Center: Mean

• Spread: Standard Deviation

• Both are not “resistant”• Both should include units

ny

y

Standard Deviation

• Takes into account how far each value in a data set is from the mean

Formula:

Find the standard deviation:10, 14, 15, 16, 20

1

2

n

yys

Properties of standard deviation1. Only use with mean2. If s = 0, there is no spread and all data pieces

are same—other wise s>0 and s gets larger as data pieces get more spread out.

3. A few outliers can really change the value of the standard deviation

Distributions with Outliers

• Tricky situation• Since outliers affect mean and standard deviation,

it is usually better to use median and IQR• However, if the distribution is unimodal—use

mean and median and just report outliers separately

• However, if you find a simple reason for outlier (input), eliminate it and use mean and standard devation—if symmetric

Other information

• If distribution is symmetric, then mean=median• If skewed right, mean>median• If skewed left, mean<median• Spread of distribution is just as important as

the center• How accurate: one or two decimal points more

than original data• Reexpress data???

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