MATH& 146

Lesson 8

Section 1.6

Averages and Variation

1

Summarizing Data

The distribution of a variable is the overall pattern

of how often the possible values occur. For

numerical variables, three summary characteristics

of the overall distribution of the data tend to be of

the most interest:

• The Average (Typical Value)

• The Variability (Spread)

• The Shape

2

Averages

Averages provide information about what is

considered the "typical" value. If you were to take

all of your data and try to reduce it to a single

number, that number would be considered the

average.

There are many ways to describe the average, but

we will focus on just three: the mean, the median,

and the mode.

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Averages

• MEAN – or arithmetic mean. It is the sum of all

values divided by the count of values. It is the most

important of the averages.

• MEDIAN – the middle value in a collection when the

values are arranged in order of increasing size. It is

the average of choice when outliers are present.

• MODE – the most common value(s) in a dataset. It

can be used for any type of data, and it is the only

average for regular categorical data.

4

Notation for the Sample

Mean

There are two ways to symbolize the sample

mean:

1) with the symbol x-bar, , or

2) with a capital letter, M. (APA notation)

x

5

The Mean

The mean of a collection of values is the sum of all

values divided by the count of values. The formula

for a mean is

where Σ x is the sum of the values and n is the

count, or sample size.

xx

n

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The Mean

The mean feels like a typical value because it is

the point where the data "balances".

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5.4 the center of gravity, or balance pointx

Example 1

Compute the mean of the following two lists of

numbers:

a) 13, 24, 25, 34, 37

b) 13, 24, 25, 34, 370

How did changing the last number from 37 to 370

affect the mean?

8

Notation for the Median

Unlike the mean, there is no standard way to

symbolize the median. Some common

abbreviations include:

1) with the abbreviation, Med (calculator notation),

or

2) with the abbreviation, Mdn (APA notation).

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The Median

The median is the middle of your data, with at

most half the data values less than it and at most

half the data more than it.

You can think of it as the value that divides the

sorted data into two equal sets of numbers.

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The Median

Let a collection of n values be written in order of

increasing size.

If n is odd, the median is the middle value in the list.

Data set 1:

24, 25, 25, 27, 29, 31, 32, 34, 37 (n = 9, odd)

29Med

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The Median

If n is even, the median is the average of the two

middle values.

Data set 2:

42, 42, 43, 44, 44, 46, 47, 47, 47, 49 (n = 10, even)

45Med

average

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Example 2

Compute the median of the following list of

numbers:

34, 13, 37, 24, 25, 13, 41, 23, 28, 31

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Example 3

Compute the median of the following two lists of

numbers:

a) 13, 24, 25, 34, 37

b) 13, 24, 25, 34, 370

How did changing the last number from 37 to 370

affect the median?

14

The Mode

A mode of a collection of values is the value (or

values) that occurs the most frequently.

For example, the set

1, 2, 2, 3, 6, 6, 6, 6, 7, 8, 10

has a mode at 6.

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The Mode

If two or more values occur equally often and more

frequently than all other values, then they each

would be considered modes.

For example, the set

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

has modes at 2 and 6

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The Mode

If no number occurs more than once, then no

mode exists.

For example,

1, 3, 5, 6, 8, 11, 12

has no mode.

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Why Modes?

The mean can only be used for numerical data,

while the median can be used for numerical and

ordinal data. The mode can be found for all data.

For instance, for the following sample of colors:

red, green, orange, orange, blue, orange

the mean and median would be impossible to find.

We can still describe the mode as the color

orange, however.

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Example 4

Find the mode of the following collection of fruit:

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Fruit Frequency

Apples 11

Oranges 12

Pears 16

Kiwis 10

Bananas 12

Example 5

Statistics exam scores for 20 students are as

follows:

50; 53; 59; 59; 63; 63; 72; 72; 72; 72; 72;

76; 78; 81; 83; 84; 84; 84; 90; 93

Find the mode.

20

Variation

In addition to describing the average, we should

also describe the variation, or spread. Measures

of variation tell us how far the numbers are

scattered about the center value of the set.

The most common ways to measure variation are

the range, interquartile range, and standard

deviation.

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Variation

• RANGE – the difference between the maximum

and minimum data values.

• INTERQUARTILE RANGE – the difference

between the upper and lower quartiles.

• STANDARD DEVIATION – the typical distance

the data values are from the mean.

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The Range

The simplest way to describe the variation of a

data set is to compute the range, defined as the

difference between the maximum and minimum

values

Although the range is easy to compute and can be

useful, it occasionally can be misleading. This is

especially true if outliers are present.

range max min

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Example 6

Consider the following two sets of quiz scores for nine students. Which set has the greater range? Would you also say that the scores in this set are more varied?

Quiz 1 Scores:

1 10 10 10 10 10 10 10 10

Quiz 2 Scores:

2 3 4 5 6 7 8 9 10

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Quartiles

Quartiles are numbers that separate the data into

quarters.

To find the quartiles, first find the median and

divide the data into two halves: the lower half are

the numbers to the left of the median and the

upper half are the numbers to the right. The

quartiles will then be the medians of each of the

halves.

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Quartiles

The lower (or first) quartile (denoted Q1) is the median of the lower half of the data. This is the point in which at most 1/4 of the values are smaller than it and at most 3/4 of the values are larger than it.

The upper (or third) quartile (denoted Q3) is the median of the upper half of the data. This is the point in which at most 3/4 of the values are smaller than it and at most 1/4 of the values are larger than it.

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Example 7

A group of eight children have the following heights

(in inches):

48, 48, 53, 53.5, 54, 60, 62, 71

Find the quartiles for the distribution of the

children's heights.

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Interquartile Range

When you are using the median to describe the

average, an appropriate measure of variation is

called the interquartile range.

The interquartile range, IQR, tells us how much

space the middle 50% of the data roughly occupy.

It is given by the formula

3 1IQR .Q Q

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Example 8

A group of eight children have the following heights

(in inches):

48, 48, 53, 53.5, 54, 60, 62, 71

Find the range and the interquartile range for the

distribution of the children's heights.

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Q1 = 50.5 Q3 = 61

Example 9

Returning to an Example 6, compute the IQR for each

set of quiz scores. Does the IQR appear to be more

reliable than the range? Why or why not?

Quiz 1 Scores:

1 10 10 10 10 10 10 10 10

Quiz 2 Scores:

2 3 4 5 6 7 8 9 10

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