Summary Statistics: Mean, Median, Standard

Deviation, and More

“Seek simplicity and then distrust it.”(Dr. Monticino)

Assignment Sheet Read Chapter 4 Homework #3: Due Wednesday Feb. 9th

Chapter 4 exercise set A: 1 -6, 8, 9 exercise set C: 1, 2, 3 exercise set D: 1 - 4, 8, exercise set E: 4, 5, 7, 8, 11, 12

Quiz #2 will be over Chapter 2 Quiz #3 on basic summary statistic calculations –

mean, median, standard deviation, IQR, SD units If you’d like a copy of notes - email me

Overview Measures of central tendency

Mean (average) Median Outliers

Measures of dispersion Standard deviation

Standard deviation units Range IQR

Review and applications

Central Tendency

Measures of central tendency - mean and median - are useful in obtaining a single number summary of a data set Mean is the arithmetic average Median is a value such that at least 50% of the

data is less and at least 50% is greater

Example

Calculate mean and median for following data sets

37 44 55 78 100 111 125 151 161

37 44 55 69 90 120 125 152 157 161

Outliers and Robustness Mean can be sensitive to outliers in data set Not robust to data collection

errors or a single unusual measurement

Blind calculation can give misleading results

mean = 170.35

median = 151

162166158154147150141233278288148152149265

212154148158150137142149148145143152

Outliers and Robustness Always a good idea to plot data in

the order that it was collected Spot outliers Identify possible data collection errors

mean without outliers = 150.14

median without outliers

= 1490

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Data

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Outliers and Robustness

Median can be a more robust measure of central tendency than mean Life expectancy

U.S. males: mean = 80.1, median = 83 U.S. females: mean = 84.3, median = 87

Household income Mean = $51,855, median = $38,885 .3% account for 12% of income

Net worth Mean = $282,500, median = $71,600

Which Central Tendency Measure?

Calculate mean, median and mode Plot data Create histogram to inspect mode(s) Do not delete data points

If analyze data without outliers, report and explain outliers

Many statistical studies involve studying the difference between population means Reporting the mean may be dictated by

objective of study

Which Central Tendency Measure?

If data is Unimodal Fairly symmetric Mean is approximately equal to median Then mean is a reasonable measure of central

tendency

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Histogram

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25 31 37 43 49 55 61 67 73

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Which Central Tendency Measure?

If data is Unimodal Asymmetric Then report both median and mean

Difference between mean and median indicates asymmetry

Median will usually be the more reasonable summary of central tendency

Histogram

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45 54 63 72 81 90 99M

ore

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0102030405060708090

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Which Central Tendency Measure?

If data is Not unimodal Then report modes and cautiously mean and

median Analyze data for differences in groups around

the modes

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Histogram

02468

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Limitations of Central Tendency

Any single number summary may not adequately represent data and may hide differences between data sets Example

2 98

50 99

100 100

150 101

198 102

Measures of Dispersion

Including an additional statistic - a measure of dispersion - can help distinguish between data sets which have similar central tendencies Range: max - min Standard deviation: root mean square difference

from the mean

n

mxmxmxmxs nn

221

22

21 )()(...)()(

Measures of Dispersion

Examples Range

1962198 498102

Measures of Dispersion Example

s Standar

d deviation

m = 100 m = 100

6.695

)100198()100150()100100()10050()1002( 22222

SD

Measures of Dispersion

Both range and standard deviation can be sensitive to outliers However, many data sets can

be characterized by mean and SD

If the values of the data set are distributed in an approximately bell shape, the ~68% of the data will be within 1

SD unit of mean, ~95% will be within 2 SD units and nearly all will be within 3 SD units -3.00 -1.00 1.00 3.00

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Measures of Dispersion Example

Suppose data set has mean = 35 and SD = 7

How many SD units away from the mean is 42?

How many SD units away from the mean is 38?

How many SD units away from the mean is 30?

Assuming bell shape distribution, ~95% are between what two values?

17

)3542(

43.7

)3538(

714.7

)3530(

49)7*235( and 21)7*235(between

Measures of Dispersion

A robust measure of dispersion is the interquartile range Q1: value such that 25% of data less than, and

75% greater than Q3: value such that 75% less than, and 25%

greater than IQR = Q3 - Q1

Example Calculate range, standard

deviation and interquartile range for the following data sets

1 98 99 100 100 100 102 102 104 107

95 98 99 100 100 100 102 102 104 107

Assignment, Discussion, Evaluation

Read Chapter 4 Discussion problems

Chapter 4 exercise set A: 1 -6, 8, 9 exercise set C: 1, 2, 3 exercise set D: 1 - 4, 8, exercise set E: 4, 5, 7, 8, 11, 12

Quiz #3 on basic summary statistic calculations – mean, median, standard deviation, IQR, SD units

Review of Definitions

Measures of central tendency Mean (average):

Median If odd number of data points, “middle” value If even number of data points, average of two

“middle” values

n

xxx n 21

Question and Examples Can mean be larger than median? Can

median be larger than mean? Give examples

Can mean be a negative number? Can the median?

The average height of three men is 69 inches. Two other men enter the room of heights 73 and 70 inches. What is the average height of all five men?

Questions and Examples The average of a data set is 30.

A value of 8 is added to each element in the data set. What is the new average?

Each element of the data set is increased by 5%. What is the new average?

Suppose that data consists of only 1’s and 0’s What does the average represent?

Application: an experiment is performed and only two outcomes can occur

Label one type of outcome 1 and the other 0

For the data set 31, 45, 72, 86, 62, 78, 50, find the median, Q1 (25th percentile) and Q3 (75th percentile)

Review of Definitions

Measures of dispersion Standard deviation =

Range = max - min IQR = Q3 - Q1

n

mxmxmxmxSD nn

221

22

21 )()(...)()(

Questions and Examples Can the SD be negative? Can the range? Can

the IQR? Can the SD equal 0? For the data set 3,1,5,2,1,6 find the SD, range

and IQR The average weight for U.S. men is 175 lbs and

the standard deviation is 20 lbs If a man weighs 190 lbs., how many standard

deviation units away from the mean weight is he? Assuming a normal (bell-shaped) distribution for

weight, ninety-five percent of U.S. men weigh between what two values?

Questions and Examples

The average of a data set is 23 and the standard deviation is 5 A value of 8 is added to each element

in the data set. What is the new standard deviation?

Each element of the data set is increased by 5%. What is the new standard deviation?  

(Dr. Monticino)