Transcript Mean, Median, Mode & Range

Research Methods
MEASURES OF CENTRAL TENDENCY
& DISPERSION
Vocabulary Review
Sum – the answer to an
addition problem.
Addend – the numbers you
added together to get the
sum.
6 + 9 = 15
Adopted from Monica Yuskaitis, 2000
Definition
Mean
Means
Average
Definition
Mean – the average of a
group of numbers.
2, 5, 2, 1, 5
Mean = 3
Formula for the Mean
Copyright © 2000 by Monica Yuskaitis
The Mean is found by evening out the numbers
2, 5, 2, 1, 5
Mean is found by evening out the numbers
2, 5, 2, 1, 5
Mean is found by evening out the numbers
2, 5, 2, 1, 5
mean = 3
How to Find the Mean of a Group of Numbers
Step 1 – Add all the numbers.
8, 10, 12, 18, 22, 26
8+10+12+18+22+26 = 96
How to Find the Mean of a Group of Numbers
Step 2 – Divide the sum by the
number of addends.
8, 10, 12, 18, 22, 26
8+10+12+18+22+26 = 96
How many addends are there?
How to Find the Mean of a Group of Numbers
Step 2 – Divide the sum by the
number of addends.
# of addends
16
6) 96
6
36
36
sum
How to Find the Mean of a Group of Numbers
The mean or average of these
numbers is 16.
8, 10, 12, 18, 22, 26
What is the mean of these numbers?
7, 10, 16
11
What is the mean of these numbers?
2, 9, 14, 27
13
What is the mean of these numbers?
1, 2, 7, 11, 19
8
What is the mean of these numbers?
26, 33, 41, 52
38
Definition
Median
is in the
Middle
Definition
Median – the middle number
in a set of ordered numbers.
1, 3, 7, 10, 13
Median = 7
How to Find the Median in a Group of Numbers
Step 1 – Arrange the numbers
in order from least to greatest.
21, 18, 24, 19, 27
18, 19, 21, 24, 27
How to Find the Median in a Group of Numbers
Step 2 – Find the middle
number.
21, 18, 24, 19, 27
18, 19, 21, 24, 27
How to Find the Median in a Group of Numbers
Step 2 – Find the middle
number.
18, 19, 21, 24, 27
This is your median number.
How to Find the Median in a Group of Numbers
 Step 3 – If there are two middle
numbers, find the mean of these two
numbers.
18, 19, 21, 25, 27, 28
How to Find the Median in a Group of Numbers
 Step 3 – If there are two middle
numbers, find the mean of these two
numbers.
21+ 25 = 46
median
23
2) 46
What is the median of these numbers?
16, 10, 7
7, 10, 16
10
What is the median of these numbers?
29, 8, 4, 11, 19
4, 8, 11, 19, 29
11
What is the median of these numbers?
31, 7, 2, 12, 14, 19
2, 7, 12, 14, 19, 31
13
12 + 14 = 26 2) 26
What is the median of these numbers?
53, 5, 81, 67, 25, 78
5, 25, 53, 67, 78, 81
60
53 + 67 = 120 2) 120
Definition
Mode
is the most
Popular
Definition
A la mode – the most
popular or that which is in
fashion.
Baseball caps are a la mode today.
Definition
Mode – the number that
appears most frequently in a
set of numbers.
1, 1, 3, 7, 10, 13
Mode = 1
How to Find the Mode in a Group of Numbers
Step 1 – Arrange the numbers
in order from least to greatest.
21, 18, 24, 19, 18
18, 18, 19, 21, 24
How to Find the Mode in a Group of Numbers
Step 2 – Find the number that
is repeated the most.
21, 18, 24, 19, 18
18, 18, 19, 21, 24
Which number is the mode?
29, 8, 4, 8, 19
4, 8, 8, 19, 29
8
Copyright © 2000 by Monica Yuskaitis
Which number is the mode?
1, 2, 2, 9, 9, 4, 9, 10
1, 2, 2, 4, 9, 9, 9, 10
9
Copyright © 2000 by Monica Yuskaitis
Which number is the mode?
22, 21, 27, 31, 21, 32
21, 21, 22, 27, 31, 32
21
Definition
Range
is the distance
Between
Definition
 Range – the difference between
the greatest and the least value
in a set of numbers.
1, 1, 3, 7, 10, 13
Range = 12
How to Find the Range in a Group of Numbers
Step 1 – Arrange the numbers
in order from least to greatest.
21, 18, 24, 19, 27
18, 19, 21, 24, 27
How to Find the Range in a Group of Numbers
Step 2 – Find the lowest and
highest numbers.
21, 18, 24, 19, 27
18, 19, 21, 24, 27
How to Find the Range in a Group of Numbers
Step 3 – Find the difference
between these 2 numbers.
18, 19, 21, 24, 27
27 – 18 = 9
The range is 9
What is the range?
29, 8, 4, 8, 19
4, 8, 8, 19, 29
29 – 4 = 25
What is the range?
22, 21, 27, 31, 21, 32
21, 21, 22, 27, 31, 32
32 – 21 = 11
What is the range?
31, 8, 3, 11, 19
3, 8, 11, 19, 31
31 – 3 = 28
What is the range?
23, 7, 9, 41, 19
7, 9, 23, 19, 41
41 – 7 = 34
the variance of a
random variable (or
somewhat more
precisely, of a
probability distribution)
is one measure of
statistical dispersion,
averaging the squared
distance of its possible
values from the expected
value.
Variance
The variance is
computed as the average
squared deviation of
each number from its
mean. For example, for
the numbers 1, 2, and 3,
the mean is 2 and the
variance is
Calculating Variance
The Variance of a
sample is given by
replacing the expected
value with the mean of
the sample
Variance in a Sample
However, the previous
equation is biased, since
it does not allow for the
fact that a parameter
(the mean) is fixed.
Hence, the unbiased
estimate of the sample
variance employs N-1 in
the denominator.
Since samples are
usually used to estimate
parameters, s² is the
most commonly used
measure of variance.
Calculating the variance
is an important part of
many statistical
applications and
analyses.
Unbiased Estimate of Sample
Variance
The Standard Deviation
divides a distribution into
standard units
represented as deviations
from the mean.
One standard deviation
away from the mean in
either direction on the
horizontal axis (the red
area on the above graph)
accounts for somewhere
around 68 percent of the
people in this group. Two
standard deviations away
from the mean (the red
and green areas) account
for roughly 95 percent of
the people. And three
standard deviations (the
red, green and blue areas)
account for about 99
percent of the people.
Standard Deviation
Suppose your data
follows the classic bell
shaped curve pattern.
One conceptual way to
think about the standard
deviation is that it is a
measures of how spread
out the bell is.
Shown here is a bell
shaped curve with a
standard deviation of 1.
Notice how tightly
concentrated the
distribution is.
Standard Deviations
Shown here is a different
bell shaped curve, one
with a standard
deviation of 2. Notice
that the curve is wider,
which implies that the
data are less
concentrated and more
spread out.
Standard Deviations
Finally, a bell shaped
curve with a standard
deviation of 3 appears
below. This curve shows
the most spread.
Standard Deviations
Let’s calculate the
standard deviation of a
set of data.
Equation for the Standard
Deviation of a Sample
The seven
The Seven values in this
data set are 73, 58, 67, 93,
33, 18, and 147. The mean
for this data set is 69.9.
For each data value,
compute the squared
deviation by subtracting
the mean and then
squaring the result.
The sum of these
squared deviations is
10,692.87. Divide by
6 to get 1782.15. Take
the square root of
this value to get the
standard deviation,
42.2.
(73-69.9)2 =
(3.1)2 =
9.61
(58-69.9)2 =
(-11.9)2 =
141.61
(67-69.9)2 =
(-2.9)2 =
8.41
(93-69.9)2 =
(23.1)2 =
533.61
(33-69.9)2 =
(-36.9)2 =
1361.61
(18-69.9)2 =
(-51.9)2 =
2693.61
(147-69.9)2 =
(77.1)2 =
5944.41
Calculating Standard Deviations
Standard Deviation
Copyright © 2000 by Monica Yuskaitis