Describing Quantitative Data Numerically

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Describing Quantitative Data Numerically
Symmetric Distributions Mean, Variance, and
Standard Deviation
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Symmetric Distributions

• Describing a typical value for a set of data
  when the distribution is at least approximately
  symmetric allows us to choose our measure of
  center
• We can use either
• Mean
• Median

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Finding the Mean of a Distribution

• The mean of a set of numbers is the arithmetic
  average. We find this value by adding together
  each value and then dividing by the number of
  values we added together
• The formula for the mean is

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Lets see the Formula in Action

• Consider Babe Ruths HR data
• A check of a dotplot indicates that the
  distribution is approximately symmetric

54 59 35 41 46 25 47 60
54 46 49 46 41 34 22
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• So the first step is to add all the values
• 54 59 35 41 46 25 47 60 54 46
  49 46 41 34 22
• 659
• Now we need to divide that sum by the number of
  values we added together.

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• So the mean of the data is 43.9333. Now, if we
  wish to talk about the typical number of home
  runs for Babe Ruth (and we ALWAYS wish to talk
  about the context of our data!), we could say
  something like
• On average, Babe Ruth hit approximately 44 home
  runs per season during the 15 seasons he played.

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• Remember that although the center is a very
  important part of our description, we also need
  to look at the spread of the distribution.
• When we use the mean as our measure of center, we
  use the standard deviation as our measure of
  spread.
• We can think of standard deviation as an average
  distance of values from the mean
• To calculate the standard deviation by hand,
  well make a data table

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Calculating Standard Deviation
S
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X X X - X (X X)2
54 43.9333 10.0667 101.3384
59 43.9333 15.0667 227.0054
35 43.9333 -8.9333 79.8038
41 43.9333 -2.9333 8.6042
46 43.9333 2.0667 4.2712
25 43.9333 -18.9333 358.4698
47 43.9333 3.0667 9.4046
60 43.9333 16.0667 258.1388
54 43.9333 10.0667 101.3384
46 43.9333 2.0667 4.2712
49 43.9333 5.0667 25.6714
46 43.9333 2.0667 4.2712
41 43.9333 -2.9333 8.6042
34 43.9333 -9.9333 98.6704
22 43.9333 -21.9333 481.0696
SUM .0005 (essentially 0) 1770.9333
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Creating the Data Table
X - X
54 43.9333 10.0667
15.0667
-8.9333
-2.9333
2.0667
-18.9333
3.0667
16.0667
10.0667
2.0667
5.0667
2.0667
-2.9333
-9.9333
-21.9333

• The first part of our formula indicates that we
  need to find the distance from the mean for each
  of our values (x x)

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• Now that we know the individual distances for
  each value, we want to find an average of those
  distances.
• To find an average we have to add all the values
  together
• We find, though, that the sum of those values is
  always zero.
• Why? Because some of the values are above the
  mean (positive values) and some are below
  (negative). The positives and negatives cancel
  each other out.
• So what values can we use to find the average
  distance from the mean for a set of values?

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• One way to get rid of the negative values in
  these distances is to square each of the values.
  Thats exactly what our formula tells us to do.
  (x x)2
• Once we have these values, to find the average we
  must add them together

(X X)2
101.3384
227.0054
79.8038
8.6042
4.2712
358.4698
9.4046
258.1388
101.3384
4.2712
25.6714
4.2712
8.6042
98.6704
481.0696
SUM 1770.9333
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• The final step in finding an average is to divide
  by the number of values we added together, but
  our formula is a little different here.

• Instead of dividing by the total number of values
  we added together, we divide by 1 less than the
  total.
• Why? We have taken a sample of the data
  instead of every piece of data in the population.
  Since another sample would produce a slightly
  different mean, it would also produce a slightly
  different standard deviation. Dividing by 1 less
  than the total number of values added together
  will give us a slightly larger spread to account
  for this sampling variation.

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• So, we divide the sum of the squared deviations
  by n-1
• We have now calculated everything inside the
  square root sign
• This value is an important oneIt is called the
• Variance --S2

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• Since the units of the variance are not the same
  as our original units, we have one more
  calculation we must make.
• The square root of the variance will restore the
  original units and give us the average distance
  from the meanthe standard deviation
• S 11.2470

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TI-TipsMean, Variance, Standard Deviation

• Find the
• MEAN
• Enter the data into a list
• 2nd STAT
• MATH
• 3mean(list name)
• If you have used a frequency list,
• 3mean(data list, freq list)

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TI-Tips

• Find the Variance
• Enter the data in a list
• 2nd STAT
• MATH
• 8variance(list name)
• If you have used a frequency list,
• 8variance(data list, freq list)

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TI-Tips

• Find
• Standard Deviation
• Enter the data in a list
• 2nd STAT
• Math
• 7stdDev(list name)
• If you have used a frequency list,
• 7stdDev(data list, freq list)

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TI-Tips

• Find
• Mean and Std Dev.
• Enter the data in a list
• STAT
• Calc
• 11-Var Stats(list name)
• Enter
• If you have used a frequency list,
• 11-var stats(data list, freq list)

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Additional Resources

• Practice of Statistics Pg 30-34, 43-46

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