BI 152 Week 12 sp. 2011
Name _____________________________
DISPERSION Ambrose Ch. 5
In Ambrose we have been working in Chapter 4, where the types of data and measures of central tendency
are described.
Last week we saw two samples of data that had the same mean, median and mode. But they were quite
different in structure. I have added a third example with a different mode.
1. What are the ranges? (see below.)
Sample A had a wide range.
Sample B had a narrow range.
But that still doesn’t tell us how the observations are clustered about the mean.
Example (I have added a third sample with same mean and median, different mode):
Sample A
Foot length
(cm)
Sample B
Foot length
(cm)
31.0
17.8
27.0
31.1
27.0
26.8 mean
27 median
27 mode
26.0
27.0
29.5
24.5
27.0
Range =
13.2
Standard
deviation
(s)
s=
26.8
27 median
27 mode
Sample C
Foot
Length
(cm)
16.
15
27
38
38
26.8
27 median
38 mode
Range =
Range =
s=
s=
How can we describe that difference in the way the numbers are spread out?
We use measures of dispersion – how spread out is the distribution? Is it close to the mean or not?.
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BI 152 Week 12 sp. 2011
Name _____________________________
Measures of dispersion
Standard deviation is most commonly used.
The most common ones are the Variance and the Standard Deviation. Variance is simply the square of the
standard deviation. So, if standard deviation is symbolized by “s” then variance is designated by s 2. Both
measure the spread of the data.
Standard Deviation Definition:
Standard deviation is a statistical measure of spread or variability.
It is the value that divides the normal distribution up so that one standard deviation
from the mean in both directions includes 68.26% of the data.
What does that mean?
All the area under the bell curve represents 100% of the observations.
The standard deviation partitions that space so
that the mean, + or – one standard deviation
includes 68.26% of the data. That’s a definition.
Distribution
A
The value of the standard deviation is applied to the X axis. It divides the distribution up into defined units.
Two standard deviations on both sides of the mean include just over 95% of the data.
And three standard deviations on both sides of the mean include 99.74% of the data.
Notice that the area under the curve becomes smaller as you move out toward the tails of the distribution.
That is why less and less of a percentage of the data is included in each standard deviation away from the
mean.
2. Which distribution has a smaller
standard deviation, A or B (assume the x axis
has the same scale in both)?
Distribution
B
3. How do you know?
4. How does standard deviation relate to accepting or rejecting a null hypothesis?
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BI 152 Week 12 sp. 2011
Name _____________________________
Calculating the Standard Deviation
The standard deviation is calculated from the data using this formula. Note: the formula in Ambrose is a
different version of this. But I like this one. You don’t need to memorize this.
The Σ symbol (sigma) means “sum of” You add up what follows the sigma symbol.
In this formula, X is an observation.
M is the mean for that sample. (We use X with a bar over it for mean.)
n = the number of observations in the sample.
So, subtract the mean from the observation. This is the “deviation” from the mean.
Then square that number. Do this for all observations.
Add up those squared deviations.
Divide the sum of those squared deviations by n-1.
Finally, take the square root of that to get the value for the standard deviation.
5. Let’s do one by hand. I have calculated the first deviation and squared it.
Sample A
Foot
length
(cm)
_
(x – x) =
_
(x – x)2=
_
Σ (x – x)2=
n-1
(31-26.8) =
4.2
31.0
17.64
17.8
27.0
31.1
27.0
Mean =
26.8
_
Σ (x – x)2=
Let’s never do that again.
You can use excel to calculate the standard deviations. Simply type =STDEV(A1:Z99) (of course you
will fill in the actual range of your data points).
You can also use http://www.graphpad.com/quickcalcs/CImean1.cfm You just type in your data for the
sample, one number per line, and it will calculate the mean and standard deviation and other descriptive
statistics.
For another explanation of standard deviation, with great illustrations, go to
http://www.mathsisfun.com/data/standard-deviation.html
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BI 152 Week 12 sp. 2011
Name _____________________________
Have you got the idea?
6. Try this.
Standard
A
B
deviation
s=2 5.413
s = 6.82
(s)
Which distribution goes with which standard deviation?
I
C
s = 11.26
II
III
7. Suppose the mean height in BI 152 was 170 cm and the standard deviation was 10 cm
How tall would a person be who is one standard deviation taller than the class average?
How tall would a person be who is two standard deviations smaller than the class average?
What is the range of heights that includes 95% of the data?
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