The document discusses various measures of central tendency and dispersion used to describe sets of data. It defines the mean, median, and mode as common measures of central tendency, and explains how to calculate each one. For measures of dispersion, it introduces the range, variance, and standard deviation, defining what each measures and providing formulas to calculate variance and standard deviation. Examples are included to demonstrate calculating various measures for sample data sets.

1. Measures of Central Tendency

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

3. 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

4. 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



5. 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.

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

7. 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

8. 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

9. 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

10. 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.

11. 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.

12. 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.

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

14. The mode is 2.

15. 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

16. Measures of Dispersion…..Arrays

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

18. 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

19. 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

20. 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

21. 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

22. 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.

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

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

25. 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.

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

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

28. 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

29. 5
.
4
4
18
4
125
143
4
5
25
143
2
2






s
1
)
( 2
2
2





n
n
x
x
s

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