copyright (c) 2002 houghton mifflin company. all rights reserved. 1 averages and variation

Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1

Averages and Variation


Today

• Check in– Quiz next Tuesday– Will NOT NOT NOT include today’s

material• So you can focus on other lectures

– Proposal draft assigned• It’s a draft of the full proposal• Due Tuesday november 22nd


Measures of Central Tendency

• Mode

• Median

• Mean


The Mode

the value or property that occurs most frequently in the data


Find the mode:

6, 7, 2, 3, 4, 6, 2, 6

The mode is 6.


Find the mode:

6, 7, 2, 3, 4, 5, 9, 8

There is no mode for this data.


The Median

the central value of an ordered distribution


To find the median of raw data:

• Order the data from smallest to largest.

• For an odd number of data values, the

median is the middle value.

• For an even number of data values, the

median is found by dividing the sum of

the two middle values by two.


Find the median:

Data: 5, 2, 7, 1, 4, 3, 2

Rearrange: 1, 2, 2, 3, 4, 5, 7

The median is 3.


Find the median:

Data: 31, 57, 12, 22, 43, 50

Rearrange: 12, 22, 31, 43, 50, 57

The median is the average of the middle two values =

372

4331


The Mean

The mean of a collection of data is found by:• summing all the entries• dividing by the number of entries

entriesofnumberentriesallofsum

mean


Find the mean:

6, 7, 2, 3, 4, 5, 2, 8

6.4625.48

378

82543276mean


Sigma Notation

•The symbol means “sum the following.”

• is the Greek letter (capital) sigma.


Notations for mean

Sample mean

“x bar”

Population mean

Greek letter (mu)x


Number of entries in a set of data

• If the data represents a sample, the

number of entries = n.

• If the data represents an entire

population, the number of entries = N.


Sample mean

nx

x


Population mean

N

x


Resistant Measure

a measure that is not influenced by extremely high or low data values


Which is less resistant?

• Mean• Median

The mean is less resistant. It can be made arbitrarily large by increasing the size of one value.


Weighted Average

Average calculated where some of the numbers are assigned more

importance or weight


Weighted Average

x. value data the ofweight the w

AverageWeighted

where

w

xw


Compute the Weighted Average:

• Midterm grade = 92• Term Paper grade = 80• Final exam grade = 88• Midterm weight = 25%• Term paper weight = 25%• Final exam weight = 50%


Compute the Weighted Average:

x w xw• Midterm 92 .25 23• Term Paper 80 .25 20• Final exam 88 .50 44

1.00 87

Average Weighted8700.1

87

w

xw


Percentiles

For any whole number P (between 1 and 99), the Pth percentile of a distribution is a value such that P% of the data fall at or below it.

The percent falling above the Pth percentile will be (100 – P)%.


Percentiles

40% of data

Low

est

valu

e

Hig

hes

t va

lueP 40

60% of data


Quartiles

• Percentiles that divide the data into fourths

• Q1 = 25th percentile

• Q2 = the median

• Q3 = 75th percentile


Computing Quartiles

• Order the data from smallest to largest.• Find the median, the second quartile.• Find the median of the data falling below

Q2. This is the first quartile.

• Find the median of the data falling above Q2. This is the third quartile.


Find the quartiles:

12 15 16 16 17 18 22 22

23 24 25 30 32 33 33 34

41 45 51

The data has been ordered.

The median is 24.


Find the quartiles:

12 15 16 16 17 18 22 22

23 24 25 30 32 33 33 34

41 45 51

For the data below the median, the median is 17.

17 is the first quartile.


Find the quartiles:

12 15 16 16 17 18 22 22

23 24 25 30 32 33 33 34

41 45 51

For the data above the median, the median is 33.

33 is the third quartile.


Find the interquartile range:

12 15 16 16 17 18 22 22

23 24 25 30 32 33 33 34

41 45 51

IQR = Q3 – Q1 = 33 – 17 = 16


Measures of Variation

• Range

• Standard Deviation

• Variance—but we won’t talk about this


The Range

the difference between the largest and smallest values of a

distribution


Find the range:

10, 13, 17, 17, 18

The range = largest minus smallest

= 18 minus 10 = 8


The standard deviation

a measure of the average variation of the data entries from the mean


Standard Deviation

• Tells us how much data entries differ from the mean

• Why do we care? Can’t we just calculate the mean and the range?


Standard Deviation—why?

• Suppose 2 data sets:• 1, 4, 4, 5, 6, 7, 8, 9, 10; range = 10-1=9• Mean = 54/9 = 6• Or• 1, 2, 5, 6, 7, 7, 7, 9, 10; range = 10-1=9• Mean = 54/9 = 6• Data sets are different, but the mean and range

are the same.


Standard Deviation

• Knowing HOW the data are arranged (distributed) tells us more than the mean and range.

• A lot of variability, or not very much variability?

• Especially important in large data sets where it may be impossible to ‘eyeball’ the variability.


Standard deviation of a sample

1n

)xx(s

2

n = sample size

mean of the sample


To calculate standard deviation of a sample

• Calculate the mean of the sample.• Find the difference between each entry (x) and the

mean. These differences will add up to zero.• Square the deviations from the mean.• Sum the squares of the deviations from the

mean.• Divide the sum by (n 1) to get the variance.• Take the square root of the variance to get

the standard deviation.


Find the standard deviation

x302622

2)x(x xx

4 04

16 016___3278 mean=

26

Sum = 0

copyright (c) 2002 houghton mifflin company. all rights reserved. 1 averages and variation

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