Download Measures of Dispersion

Measures of Dispersion
Section 4.3
The case of Fred and Barney at the
bowling alley
• Fred and Barney are at the bowling alley and
they want to know who’s the better bowler.
• They bowl six games and here are the results:
Fred
185 135 200 185 155 250
Barney 182 185 188 185 180 190
Find the average, Find the median,
Find the mode!!!
• After the games it’s time for Fred and Barney
to do the math.
• We compute Fred’s mean and we see it is 185.
• Barney’s mean is computed and it is also 185.
• We look at Fred’s median it’s 185.
• Barney’s median is also 185.
• The mode for Fred is 185 and the mode for
Barney is 185.
• Make sure you can do these calculations.
Mr. Consistency vs. Mr. Loose
Cannon
• If we look at the scores we notice that Barney’s
scores are very consistent. They do not vary
much around his average.
• Fred however has wildly varying scores. His last
game was 250, so maybe he’s Mr. Clutch.
• Statistically speaking we would like to measure
this variation about the mean.
• What we need to do is to calculate deviations
from the mean, sample variance, and sample
standard deviation.
Deviation from the mean
• First let’s calculate the deviation from the mean for each
score.
• The formula is d i  x  x .
• For Fred the avg. of his scores is 185.
• d1 = 185 – 185 = 0
• d2 = 135 – 185 = -50
• d3 = 200 – 185 = 15
• d4 = 185 – 185 = 0
• d5 = 155 – 185 = -30
• d6 = 250 – 185 = 65
• The sum of the deviations is zero! This is always true.
• What we need to do is get rid of those pesky minus
signs.
The sample variance
• The measurement we need is called the
sample variance.
• What we do is we square each deviation
and then sum them up and divide by one
less the number of data points.
• The formula is give as:
1
1
2
2
s 
(
x

x
)

d


i
i
n 1 i
n 1 i
2
The Sample variance for Fred
• Lets calculate the sample variance for
Fred.
1
s 
(0 2  (50) 2  (15) 2  (0) 2  (30) 2  (65) 2 )
6 1
1
2
s  (0  2500  225  0  900  4225)
5
7850
2
s 
 1570
5
2
The standard deviation.
• Since we want an understanding of how the data
is dispersed about the mean, then the statistic
that measures this must be of the same units as
the mean.
• Unfortunately the sample variance is the square
of these units. So what we should do is take the
square root of the sample variance.
• This is called the sample standard deviation.
• Sample Standard Deviation = sample variance
The standard deviation for Fred
• We can now calculate Fred’s sample
standard deviation.
s  1570  39.62
• You should calculate Barney’s sample
variance and sample standard deviation.
2
• They are s  13.6
s  13.6  3.69
One Standard deviation from the
mean
• Sometimes it is useful to compute the percentage of the
data that is one standard deviation from the mean.
• What you need to do is first compute the mean of the
sample.
• Then compute the standard deviation of the sample.
• Next, you want to construct the interval that is one
standard deviation from the mean.
• The left end point is x  s.
• The right end point is x  s.
• Find the number of data points that fall into this range
and then divide by the total number of data points.
• When this number is close to 68%, this is indicative of a
normal distribution.