Frequency Distributions and Histograms

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Frequency Distributions and Histograms

• Statistics Section 2.2

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Histograms

• Look like bar graphs but also have the following
  criteria
• The bars have the same width and always touch
• The width of a bar represents a quantitative
  value, such as age, rather than a category
• The height of each bar indicates frequency
• They give information about a range of
  individuals not just a single individual.

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Histograms

• Your first objective is to figure out how many
  bars (or classes) you want. Usually 5 to 15
  groups are used.
• Next, find a class width.
• (range)/number of classes
• Always increase to the next whole number, even if
  you got a whole number as an answer.

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Class Limits

• The lower class limit is the lowest value in a
  particular class, as the upper class limit is the
  highest value in a particular class.
• The class width is the difference between the
  lower class limit of one class with the lower
  class limit of the next class.

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Midpoint (class mark)

• The center of the class
• (lower class limit) (upper class limit)/2

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Frequency Table

• A frequency table lists the following
• The limits of each class
• The frequency with which the data fell into a
  class
• The class midpoint

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Class Boundaries

• We dont want a space between the bars, so we
  meet halfway between the difference of the
  lower-upper limit and the higher-lower limit.

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Example Commuting Distance in Dallas
13 47 10 3 16 20 17 40 4 2
7 25 8 21 19 15 3 17 14 6
12 45 1 8 4 16 11 18 23 12
6 2 14 13 7 15 46 12 9 18
34 13 41 28 36 17 24 27 29 9
14 26 10 24 37 31 8 16 12 16
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Example Step 1 Class Width

• I want there to be six classes bars eventually.
• (largest value)-(smallest value))/Number of
  classes
• If I want 6 classes
• (47-1)/6 7.7 -gt 8
• So my class width is 8.

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Example Step 2 Class Limits

• Determine the lower limits
• Smallest value is 1 and my class width is 8.
• So my lower class limits are 1, 9, 17, 25, 33, 41
• Determine the upper limits
• The second class begins at 9 so my upper limit
  for my first class must be 8.
• 8, 16, 24, 32, 40, 48

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Example Step 3 Find Midpoints

• The center of the class
• (lower class limit) (upper class limit)/2
• (18)/2 9/2 4.5
• (916)/2 25/2 12.5
• (1724)/2 41/2 20.5
• 28.5
• 36.5
• 44.5

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Example Step 4 Find the Class Boundaries

• Extend your class limits by ½ both ways.
• Class 1 0.5 8.5
• Class 2 8.5 16.5
• Class 3 16.5 24.5
• Class 4 24.5 32.5
• Class 5 32.5 40.5
• Class 6 40.5 48.5

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Example Step 5 Create a tally

• Count how many people fall into each class and
  create a bar graph from that.

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Relative-frequency Histograms

• Relative frequency Class frequency / total of
  all frequencies
• Percentages
• The graphs should look the same except the
  vertical scales will be different