Measrue of central tendency

1. Measures of Central Tendency

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

3. Relevance
To be able to calculate the
most appropriate measure of
center after analyzing the
context of a study that might
or might not contain extreme
values.

4. 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

5. 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



6. 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.

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

8. 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

9. 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

10. 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

11. Example 2 – Use GDC
Find the mean of the array.
2.0, 4.9, 6.5, 2.1, 5.1, 3.2, 16.6
Use your lists on the calculator
and follow the steps.

12. Stat, Edit – input list

13. Stat, Calc, One-Var Stats, L1

14. Or…..(I like this way better!)
Home: 2nd
Stat
Math
3: Mean (L#)

15. Rounding
The mean (x-bar) is 5.77.
We used 2 decimal places
because our original data had 1
decimal place.

16. 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.

17. 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.

18. Ex 2 – Use Calculator
Input data into L1.

19. Run “Stat, Calc, One-Variable
Stats, L1”
Cursor all the way down to
find “med”

20. Or…….from the home screen
2nd
Stat
Math
4: Median(L#)

21. 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.

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

23. The mode is 2.

24. 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

25. Midrange
The number exactly midway
between the lowest value and
highest value of the data set.
It is found by averaging the low
and high numbers.
2
)
( Value
High
value
Low
midrange



26. Example
Find the midrange of the set.
3, 3, 5, 6, 8
5
.
5
2
11
2
)
8
3
(




midrange

27. Trimmed Mean

28. Trimmed Mean
 We have seen 4 different averages: the
mean, median, mode, and midrange.
For later work, the mean is the most
important.
 However, a disadvantage of the mean is
that it can be affected by extremely high
or low values.
 One way to make the mean more
resistant to exceptional values and still
sensitive to specific data values is to do

29. How to Compute a 5%
Trimmed Mean
Order the data from smallest to
largest.
Delete the bottom 5% of the data
and the top 5% of the data. (NOTE:
If 5% is a decimal round to the
nearest integer)
Compute the mean of the remaining

30. Example 80
80
50
50
42
40
40
35
35
35
30
30
30
25
23
20
20
20
20
14
a) Compute the mean for the entire sample.
b) Compute a 5% trimmed mean.
0
.
36
20
719




n
x
x
   
.
80
14
.
1
1
05
.
0
20
:
%
5
AND
REMOVE
SET
THE
OF
BOTTOM
AND
TOP
THE
FROM
VALUE
REMOVE
Trim 
7
.
34
18
625




n
x
x

31. Example 80
80
50
50
42
40
40
35
35
35
30
30
30
25
23
20
20
20
20
14
c) Compute the median for the entire sample.
d) Compute a 5% trimmed median.
5
.
32
2
35
30



Median
The median is still 32.5.
e) Is the trimmed mean or the original mean closer to the median?
Trimmed Mean

32. Weighted Average

33. Sometimes we wish to average
numbers, but we want to assign
more importance, or weight, to
some of the numbers.
The average you need is the
weighted average.

34. Formula for Weighted Average
w
xw
Average
Weighted



.
.
values
data
all
over
taken
is
sum
The
value
data
that
to
assigned
weight
the
is
w
and
value
data
a
is
x
where

35. Example:
Suppose your midterm test score is 83
and your final exam score is 95.
Using weights of 40% for the midterm
and 60% for the final exam, compute
The weighted average of your scores.
If the minimum average for an A is
90, will you earn an A?
     
2
.
90
1
57
32
60
.
0
40
.
0
60
.
0
95
40
.
0
83






Average
Weighted You will earn
an A!

36. Measures of Dispersion…..Arrays

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

38. 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

39. 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

40. 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

41. 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

42. 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.

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

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

45. 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.

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

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

48. 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

49. 5
.
4
4
18
4
125
143
4
5
25
143
2
2






s
1
)
( 2
2
2





n
n
x
x
s

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

51. Same Example – Use
Calculator
Put numbers in L1.

52. Run “Stat, Calc, One-Variable Stats,
L1” and read the numbers. Remember
you have to square the standard
deviation to get variance.
Square this number
to get the variance!

53. Or….from the home screen
2nd Stat
Math
7:stdDev(L1)
Enter

54. Variance – Using Formula
Square the ENTIRE number for
the standard deviation not the
rounded version you gave for
your answer.
5
.
4
)
121320344
.
2
( 2
2


s

55. Variance using GDC
2nd Stat
Math
8: Variance (L1)