Download measure of central tendency

Part 3 Module 2
Measures of Central Tendency
To paraphrase Benjamin Disraeli:
"There are lies, darn lies, and DAM
STATISTICS."
Compute the measures of central tendency
for the following DAM STATISTICS.
Dam Statistics
Measures of Central Tendency
A measure of central tendency is a number that
represents the typical value in a collection of
numbers. Three familiar measures of central
tendency are the mean, the median, and the
mode.
We have the following collection of seven data
points:
756, 726, 710, 568, 564, 440, 440
We say n = 7 for this distribution.
The Mean
data: 756, 726, 710, 568, 564, 440, 440
In any collection of n data points, the mean is the
sum of all n data points, divided by n.
For the dam statistics:
MEAN = (756 + 726 + 710 + 568 + 564 + 440 + 440)/7
= 4204/7
= 600.57 after rounding
The Median
In any collection of n data points, the median is "middle"
data point (or average of two middle data points) when
the data points are arranged in numerical order.
In the case of the dam statistics, the data points are
already in numerical order.
756, 726, 710, 568, 564, 440, 440
Now we choose the number in the middle of the list.
756, 726, 710, 568, 564, 440, 440
The median is 568.
The Median
When calculating the median, it is important to put the data in
numerical order, first.
If we have an odd number of data points, then there will be one
number in the middle of the ordered list. That number is the
median.
If we have an even number of data points, then there will be two
numbers in the middle of the ordered list. The median will be the
average of those two middle numbers.
The Mode
In any collection of numerical data points, the mode is the
number that occurs most often, if there is such a
number.
A distribution of numerical data points will always have a
mean or median, but may not have a mode.
In the case of the dam statistics, the number 440 occurs
more often than the other numbers, so the mode is
440.
756, 726, 710, 568, 564, 440, 440
Measures of central tendency
We would like to be able to calculate the measures of
central tendency for distributions that include more
than just a handful of data points.
Example
Find the mean, median and mode for the following
collection of responses to the question: "How many
parking tickets have you received this semester?"
1, 1, 0,1, 2, 2, 0, 0, 0, 3, 3, 0, 3, 3, 0, 2, 2, 2, 1, 1, 4, 1, 1, 0, 3, 0, 0, 0,
1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1,
1, 1, 3, 3, 0, 3, 3, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 3, 3, 3, 2, 3, 3, 1, 1, 1, 2, 2,
2, 4, 5, 5, 4, 4, 1, 1, 1, 4, 1, 1, 1, 3, 3, 5, 3, 3, 3, 2,3, 3, 0, 0, 0, 0, 3, 3,
3, 3, 3, 3, 0, 2, 2, 2, 2, 1, 1, 1,3, 1, 0, 0, 0,1, 1, 3, 1, 1, 1, 2, 2, 2, 4,
2, 2, 2, 1, 1, 1, 1,0, 0, 2, 2, 3, 3, 2, 2, 3, 2, 0, 0, 1, 1, 3, 3, 3, 1, 1, 1, 1,
1, 2, 2, 2, 2, 1, 1, 1, 1, 0,1, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3,
3, 5, 3, 3, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 4, 1, 1, 4, 4, 4, 4, 4, 4, 1, 1, 1, 2, 2,
5, 5, 2, 3, 3, 4, 4, 3, 2, 2, 2, 1, 5, 1,2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 0, 1,
1, 1, 3, 3, 3, 3, 3
Solution
It will be easy to work with this collection of data if we
organize it first. We will arrange the data numerically.
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
5, 5, 5, 5, 5, 5, 5
Summary
The value "0" appears 27 times.
The value "1" appears 96 times.
The value "2" appears 58 times.
The value "3" appears 54 times.
The value "4" appears 18 times.
The value "5" appears 7 times.
This summary provides all the information
necessary to compute the mean, median and
mode.
Frequency Tables
The summary of data from the previous slide can
be further condensed into the following
frequency table.
Frequency Tables
A frequency table always represents a list of numbers.
The numbers from the list are shown in the “value” column.
The corresponding “frequency” shows how many times a
number occurs on the list.
Find n from a frequency table
To find either the mean or the median, we
need to first find n, the number of data
points in the distribution.
To find n, we add the frequencies.
Summary: Mode, mean, median, from a
frequency table
For data in a frequency table:
Mode: the value with the greatest frequency (if there is
such a value).
Mean = S/n where n is the sum of the frequencies, and S
is the sum obtained by multiplying each frequency by its
value, and adding all those terms.
Median: Use (n+1)/2 to find the middle position. Count
frequencies until the sum of frequencies reaches or
exceeds (n+1)/2; that will show which value or values
occupy the middle position(s).
Example: Read a frequency table
A number of couch potatoes were asked 'How many hours of Nintendo did you
play yesterday?'
The responses are summarized in the table below.
Value
0
1
3
5
Frequency
14
27
17
18
Select the statement that is true.
A. Exactly 3 couch potatoes played exactly 17 hours each.
B. A total of 9 couch potatoes were surveyed.
C. Exactly 41 couch potatoes played fewer than 3 hours each.
D. None of these is true.
Calculate mean from frequency table
A number of couch potatoes were asked 'How many hours of Nintendo did you
play yesterday?'
The responses are summarized in the table below. Find the mean.
Value
0
1
3
5
A. 2.25
B. 2.21
C. 1.99
D. 2.39
Frequency
14
27
17
18
Find median from frequency table
A number of couch potatoes were asked 'How many hours of Nintendo did you
play yesterday?'
The responses are summarized in the table below. Find the median.
Value
0
1
3
5
A. 1.00
B. 2.00
C. 3.00
D. 44.50
E. 38.50
Frequency
14
27
17
18