Measures of Dispersion

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Measures of Dispersion

• Week 3

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What is dispersion?

• Dispersion is how the data is spread out, or
  dispersed from the mean.
• The smaller the dispersion values, the more
  consistent the data.
• The larger the dispersion values, the more spread
  out the data values are. This means that the
  data is not as consistent.

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Consider these sets of data

• Grades from Test 1
• - 81,83,83,82,86,81,87,80,81,86
• Grades from Test 2
• - 95,74,65,90,87,97,60,81,99,76
• What differences do you see between the two sets?
• What are the Mean scores? Ranges?
• Do you believe these grades tell a story?

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Important Symbols to remember

• X an individual value
• N Population size
• n sample population size
• i 1st data value in population

mean
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Variance

• The average of the squares of each difference of
  a data value and the mean.

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Standard Deviation

• is the measure of the average distance between
  individual data points and their mean.
• It is the square root of the variance.
• The lower case Greek letter sigma is used to
  denote standard deviation.

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How to Calculate Standard Deviation

• Given the data set 5, 6, 8, 9, calculate the
  standard deviation.
• Step 1 find the mean of the data set

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How to Calculate Standard Deviation

• Step 2 Find the difference between each data
  point and the mean.

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How to Calculate Standard Deviation

• Step 3 Square the difference between each data
  point and the mean.

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How to Calculate Standard Deviation

• Step 4 Sum the squares of the differences
  between each data point and the mean.

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How to Calculate Standard Deviation

• Step 5 Take the square root of the sum of the
  squares of the differences divided by the total
  number of data points

The average distance between individual data
points and the mean is 1.58113883 units from 7
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Standard Deviation

• Formula of what we just did
• For sample S.D. use 1/(n-1)

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When to use Pop. vs. Sample

• When we have the actual entire population (for
  example our class, 29 students), we would use the
  Population formula.
• If the problem tells us to use a particular
  formula Pop. v. Samp.
• If we are working with less entire population of
  a much larger group, we will use the sample
  formula.
• (Which is one taken away from the pop. total)

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Why is this useful?

• It provides clues as to how representative the
  mean is of the individual data points.
• For example, consider the following two data sets
  with the same means, but different standard
  deviations.

The mean with the standard deviation provides a
better description of the data set.
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TI-83 to Calculate Standard Deviation.

• Step 1. Press STAT,EDIT,1EDIT
• Step 2. Enter your data in the L1 column,
  pressing enter after every data entry.
• Step 3. Press STAT, CALC,1-Var stats
• Step 4. Scroll down to the lower case symbol for
  the Greek letter sigma
• calculator help.

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Lets try one more by hand

• Find the population standard deviation for the
  following Stats class test grades
• 78, 84, 88, 92, 68, 82, 92, 72, 88, 86, 76, 90
• (a) How many grades fall within one SD of the
  mean?
• (b) What percent fall within one SD of the mean?
• Now check it with the calculator!