Powerpoint on Central Tendency and Dispersion.ppt

Objective
To learn how to find
measures of central
tendency in a set of
raw data.

Central Values – Many times one number is used to describe the
entire sample or population. Such a number is called an average.
There are many ways to compute an average.
 There are 4 values that are considered measures of the
center.
1. Mean
2. Median
3. Mode
4. Midrange

Arrays
 Mean – the arithmetic average with which you are the
most familiar.
 Mean:
x
of
number
x
all
of
sum
bar
x 

n
x
x



Sample and Population Symbols
As we progress in this course there will be
different symbols that represent the same
thing. The only difference is that one
comes from a sample and one comes from a
population.

Symbols for Mean
Sample Mean:
Population Mean:
x


Rounding Rule
Round answers to one decimal place
more than the number of decimal
places in the original data.
Example: 2, 3, 4, 5, 6, 8
A Sample answer would be 4.1

Example
Find the mean of the array.
4, 3, 8, 9, 1, 7, 12
3
.
6
29
.
6
7
44
7
12
7
1
9
8
3
4












n
x
x

Example…….
Find the mean of the following
numbers.
23, 25, 26, 29, 39, 42, 50
4
.
33
7
234
7
50
42
39
29
26
25
23











x
n
x
x

Median
Median – the middle number in an
ordered set of numbers. Divides the
data into two equal parts.
Odd # in set: falls exactly on the middle
number.
Even # in set: falls in between the two
middle values in the set; find the average
of the two middle values.

Example
Find the median.
A. 2, 3, 4, 7, 8 - the median is 4.
B. 6, 7, 8, 9, 9, 10
median = (8+9)/2 = 8.5.

Mode
The number that occurs most often.
Suggestion: Sort the numbers in L1 to
make it easier to see the grouping of the
numbers.
You can have a single number for the mode,
no mode, or more than one number.

Example
Find the mode.
1, 2, 1, 2, 2, 2, 1, 3, 3
Put numbers in L1 and sort to see the
groupings easier.

Ex 2
Find the mode.
A. 0, 1, 2, 3, 4 - no mode
B. 4, 4, 6, 7, 8, 9, 6, 9 - 4 ,6, and 9

Measures of Dispersion…..Arrays

Dispersion
The measure of the spread or variability
No Variability – No Dispersion

Measures of Variation
There are 3 values used to measure the
amount of dispersion or variation. (The
spread of the group)
1. Range
2. Variance
3. Standard Deviation

Why is it Important?
You want to choose the best brand of
paint for your house. You are interested
in how long the paint lasts before it fades
and you must repaint. The choices are
narrowed down to 2 different paints.
The results are shown in the chart.
Which paint would you choose?

The chart
indicates the
number of
months a
paint lasts
before
fading.
Paint A Paint B
10 35
60 45
50 30
30 35
40 40
20 25
210 210

Does the Average Help?
Paint A: Avg = 210/6 = 35 months
Paint B: Avg = 210/6 = 35 months
They both last 35 months before fading.
No help in deciding which to buy.

Consider the Spread
Paint A: Spread = 60 – 10 = 50 months
Paint B: Spread = 45 – 25 = 20 months
Paint B has a smaller variance which means
that it performs more consistently. Choose
paint B.

Range
The range is the difference between
the lowest value in the set and the
highest value in the set.
Range = High # - Low #

Example
Find the range of the data set.
40, 30, 15, 2, 100, 37, 24, 99
Range = 100 – 2 = 98

Deviation from the Mean
 A deviation from the mean, x – x bar, is the
difference between the value of x and the mean x
bar.
We base our formulas for variance and standard
deviation on the amount that they deviate from
the mean.
 We’ll use a shortcut formula – not in book.

Variance (Array)
 Variance Formula
1
)
( 2
2
2





n
n
x
x
s

Standard Deviation
The standard deviation is the square
root of the variance.
2
s
s 

Example – Using Formula
Find the variance.
6, 3, 8, 5, 3
6 36
3 9
8 64
5 25
3 9
x 2
x
25

 x 143
2

 x

5
.
4
4
18
4
125
143
4
5
25
143
2
2






s
1
)
( 2
2
2





n
n
x
x
s

Find the standard deviation
The standard deviation is the square
root of the variance.
12
.
2
5
.
4 

s

Powerpoint on Central Tendency and Dispersion.ppt

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