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2-6 Measures
of Spread
Remember from yesterday, the
mean, median, and mode are
valuable to us in a couple of ways
1 They help us get a sense for the typical
value a set of data will take.
2. They allow us to reduce a great number of
values down to a single value.
2. M of CTs are easy to calculate.
There are limitations however
that we have to deal with….
Find the mean of the following
two sets of data:
Set 1: 1,2,3,4,5
1+2+3+4+5
X=
5
= 15
5
=3
Set 2: -44,-7,15,22,29
- 44 - 7 + 15 + 22 + 29
X=
5
= 15
The means are the same for
5
these 2 very different sets of
=3
data…..hmmm
Weather
As a simple example, consider average
temperatures for cities. While two cities may
each have an average temperature of 15 °C, it's
helpful to understand that the range for cities
near the coast is smaller than for cities inland,
which clarifies that, while the average is similar,
the chance for variation is greater inland than
near the coast.
So, an average of 15 occurs for one city with
highs of 25 °C and lows of 5 °C, and also occurs
for another city with highs of 18 and lows of 12.
The standard deviation allows us to recognize
that the average for the city with the wider
variation, and thus a higher standard deviation,
will not offer as reliable a prediction of
temperature as the city with the smaller variation
and lower standard deviation.
While the mean is useful for
determining the “middle” of
a set of data, we need a way
to distinguish between
various sets of data.
1,2,3,4,5
-44, -7, 15, 22, 29
A measure that allows us to
examine the difference
between these data sets is
called the spread.
A more common variable to use
to measure the spread of a set
of data is the
Standard Deviation
We need a predefinition to
understand Standard
Deviation.
Variance
A measure of dispersion
that is found by averaging
the squares of the
deviation of each piece of
data.
Standard Deviation
A measure of dispersion
found by taking the square
root of the variance.
The square root brings the
scale of the measure back
down to the scale of the
raw data…
Do not copy this page
=
(x –
2
u)
n
Population Standard Deviation
s=
(x –
2
x)
n-1
Sample Standard Deviation
For our standard deviation
calculations, we will use the
sample standard deviation
version.
Calculate the SD for the data sets :
1,2,3,4,5 and -44,-7, 15, 22, 29
Data (x)
(x – x)
(x – x)2
1
-2
4
2
-1
1
3
0
0
4
1
1
5
2
4
Subtract each from the
mean
Square each
These are the data values
(x –
2
x) =4 + 1 + 0 + 1 + 4
=10
Add up all the
squared values
10 = 10
n-1
4
= 2.5 (variance)
2.5 = 1.58 (standard
deviation)
Calculate the SD for the data sets :
1,2,3,4,5 and -44,-7, 15, 22, 29
Data (x)
(x – x)
(x – x)2
-44
-47
2209
-7
-10
100
15
12
144
22
19
361
29
26
676
(x –
2
x) = 2209 + 100 + 144
+ 361 + 676
= 3490
3490 = 3490
n-1
4
= 872.5 (variance)
872.5 = 29.54 (standard
deviation)
1,2,3,4,5
s = 1.58
-44, -7, 15, 22, 29
s = 29.54
From this we can see how the
greater the standard deviation the
greater the spread of the data.
Homework