Download C.U.S.S. Center Ppt

How to describe a
graph
Otherwise called
CUSS
1. Center
discuss where the middle of the
data falls
three types of central tendency
mean, median, & mode
Mean, Median and
Mode
Measures of Central Tendency
►Median
- the middle of the data; 50th
percentile
 Observations must be in numerical
order
 Is the middle single value if n is
odd
 The average of the middle two
values if n is even
NOTE: n denotes the sample size
Measures of Central Tendency
parameter
►Mean - the arithmetic average
 Use m to represent a population
statistic
mean
 Use x to represent a sample mean
Formula:
x
x
n
S is the capital Greek
letter sigma – it means to
sum the values that follow
Measures of Central Tendency
►Mode
– the observation that occurs the
most often
 Can be more than one mode
 If all values occur only once – there
is no mode
 Not used as often as mean & median
Suppose we are interested in the number of
lollipops that are bought at a certain store. A
sample of 5 customers buys the following number
of lollipops. Find the median.
The numbers are in order
& n is odd – so find the
middle observation.
2
The median is 4
lollipops!
3 4 8 12
Suppose we have sample of 6 customers that buy
the following number of lollipops. The median is …
The median is 5
The numbers are in order
lollipops!
& n is even – so find the
middle two observations.
Now, average these two values.
2
5
3 4 6 8 12
Suppose we have sample of 6 customers that buy
the following number of lollipops. Find the mean.
To find the mean number of lollipops
add the observations and divide by
n.
x  5.833
2  3  4  6  8  12
6
2
3 4 6 8 12
Using the calculator . . .
What would happen to the median & mean if
the 12 lollipops changed to 20?
The median is . . .
The mean is . . .
5
7.17
2  3  4  6  8  20
6 What happened?
2
3 4 6 8 20
What would happen to the median & mean if
the 20 lollipops were 50?
The median is . . .
The mean is . . .
5
12.17
2  3  4  6  8  50
6 What happened?
2
3 4 6 8 50
Resistant ►Statistics
outliers
that are not affected by
►Is
the median resistant?
►Is
the mean resistant?
YES
NO
Look at the following data set. Find the
mean & median.
Mean = 27
Median = 27
21
27
Create a histogram with the
data.
x-scale
of 2) Then
Look(use
at the
placement
of
find
mean
median.
thethe
mean
andand
median
in
this symmetrical
distribution.
23
23
24
25
25
27
27
28
30
30
26
26
26
27
30
31
32
32
Look at the following data set. Find the
mean & median.
Mean = 28.176
Median = 25
Create a histogram with the
data.
x-scale
of 8) Then
Look(use
at the
placement
of
find
mean
median.
thethe
mean
andand
median
in
this right (positive)
skewed
22
29
28 distribution.
22
24
25
28
21
23
62
23
24
23
26
36
38
25
Look at the following data set. Find the
mean & median.
Mean = 54.588
Median = 58
Create a histogram with the
data.
Then
findplacement
the meanof
and
Look
at the
median.
the mean
and median in
this skewed left (negative)
distribution.
21
46
54
47
53
60
55
55
56
63
64
58
58
58
58
62
60
Recap:
►In
a symmetrical distribution, the mean
and median are equal.
►In a skewed distribution, the mean is
pulled in the direction of the skewness.
►In a symmetrical distribution, you
should report the mean!
►In a skewed distribution, the median
should be reported as the measure of
center!
Trimmed mean:
To calculate a trimmed mean:
►Multiply the trim % (amount to trim)
by n
►Truncate that many observations from
BOTH ends of the distribution (when
listed in order)
►Calculate the mean with the shortened
data set
Find a 10% trimmed mean with the following data.
12
14
19
20
22
24
25
26
26
10%(10) = 1
So remove one observation
from each side!
14  19  20  22  24  25  26  26
 22
8
35