Variance and Standard Deviation 3 Frequency Distributions

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Variance and Standard Deviation (3)Frequency
Distributions
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Standard Deviation
Standard Deviation can more conveniently be
written
this makes manual calculations much simpler
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Frequency Distributions
Visits to the doctors 0 1 5 2 1 5 3 3 2 4 3 6 2
3 1 0 1 0 3 2
Mean 47 / 20 2.35
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Ruler Experiment
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Ruler Experiment - Mean
Estimate of Mean total (based on mean) ? total
frequency
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Mean within Frequency Distributions
Within frequency distribution, mean is defined as
...
fi means frequency
Where data is provided in ranges, the xi value
are the mid-point in the range. It represents an
estimate of the mean, since it assumes that
values are evenly distributed in the range
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Standard Deviation with Frequency Distributions
Previously, we arrived at the formula
fi means frequency
The ? xi2 part can also be calculated from the
tables
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Standard Deviation with Frequency Distributions
Previously, we arrived at the formula
with frequency distribution, it becomes
fi means frequency
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Boys
? xi2fi 24662.5
n 90
sx 9.44
? xi2fi 22087.5
Girls
n 70
sx 8.31
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Mean 47 / 20 2.35
? (8.35 5.5225) ? 2.8275 1.68
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Activity

• Page 31 of your Statistics 1 book and try
• E3

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The right average?
In a 5 person office The boss makes 50K The 2
secretaries make 14K The sales rep makes
25K The trainee sales rep gets 16K
The median pay is 14, 14, 16, 25, 50
16K
The modal pay is 14K
The mean pay is 119K ? 5 23.8K
which represents the best average ?
The boss says on average you earn over 23K in my
office
The sales rep says on average you only get 16K
in my office
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Suppose this had been our experiment
Mean, median, spread?
Cannot calculate a mean and standard deviation,
since not all data value are known
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You can still estimate the median and
inter-quartile ranges
90 boys tested 70 girls tested
Median boy 11 cm Median girl 17 cm
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You can still estimate the median and
inter-quartile ranges
Boy IQR 20 - 6 14 cm Girl IQR 25 - 11 14
cm
90 boys tested 70 girls tested
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Pros and Cons of different averages (mean and
median) and measures of spread (inter-quartile
range and standard deviation)

• Median and inter-quartile range are unaffected by
  extreme values
• therefore the most suitable measures when extreme
  value occur
• Median and inter-quartile range can be calculated
  with some data missing (in the end ranges)
• Mean and standard deviation include all values
• Mean and standard deviation are more sensitive
  measures
• they provide a better picture of the whole data
• You can therefore chose the values that bias the
  interpretation in you favour!

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"There are three kinds of lies lies, damned
lies and statistics. Mark Twain
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Activity

• Page 34 of your Statistics 1 book and try
• F3
• page 36
• Test yourself

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Median and IQR are unaffected by a change in the
upper range
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Estimate of mean 4940 / 80 61.75 sec
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951.9
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Mean and SD are changed slightly by a change in
the upper range
952.4 (951.9)