Frequency Distributions Section 2.1 Frequency Distributions and Their Graphs Larson/Farber 4th ed.

Section 2.1 Objectives Construct frequency distributions Construct frequency histograms, frequency polygons, relative frequency histograms, and ogives Larson/Farber 4th ed.

Frequency Distribution A table that shows classes or intervals of data with a count of the number of entries in each class. The frequency, f, of a class is the number of data entries in the class. Class Frequency, f 1 – 5 5 6 – 10 8 11 – 15 6 16 – 20 21 – 25 26 – 30 4 Class width 6 – 1 = 5 Lower class limits Upper class limits Larson/Farber 4th ed.

Constructing a Frequency Distribution Decide on the number of classes. Usually between 5 and 20; otherwise, it may be difficult to detect any patterns. Find the class width. Determine the range of the data. Divide the range by the number of classes. Round up to the next convenient number. Larson/Farber 4th ed.

Constructing a Frequency Distribution Find the class limits. You can use the minimum data entry as the lower limit of the first class. Find the remaining lower limits (add the class width to the lower limit of the preceding class). Find the upper limit of the first class. Remember that classes cannot overlap. Find the remaining upper class limits. Larson/Farber 4th ed.

Constructing a Frequency Distribution Make a tally mark for each data entry in the row of the appropriate class. Count the tally marks to find the total frequency f for each class. Larson/Farber 4th ed.

Example: Constructing a Frequency Distribution The following sample data set lists the amount spent on books for a semester. Construct a frequency distribution that has seven classes. 91 472 279 249 530 376 188 341 266 199 142 273 189 130 489 248 101 375 486 190 398 269 43 30 127 354 84 Larson/Farber 4th ed.

Solution: Constructing a Frequency Distribution 91 472 279 249 530 376 188 341 266 199 142 273 189 130 489 248 101 375 486 190 398 269 43 30 127 354 84 Number of classes = 7 (given) Find the class width Round up to 72

Solution: Constructing a Frequency Distribution Use 30 (minimum value) as first lower limit. Add the class width of 72 to get the lower limit of the next class. 30 + 72 = 102 Find the remaining lower limits. Lower limit Upper limit 30 102 174 246 318 390 462 Class width = 72

Solution: Constructing a Frequency Distribution The upper limit of the first class is 101 (one less than the lower limit of the second class). Add the class width of 72 to get the upper limit of the next class. 101 + 72 = 173 Find the remaining upper limits. Lower limit Upper limit 30 101 102 173 174 245 246 317 318 389 390 461 462 533 Class width = 72 Larson/Farber 4th ed.

Solution: Constructing a Frequency Distribution Make a tally mark for each data entry in the row of the appropriate class. Count the tally marks to find the total frequency f for each class. Class Tally Frequency, f 30 - 101 lllll 5 102 - 173 lll 3 174 - 245 246 - 317 lllll ll 7 318 - 389 llll 4 390 - 461 l 1 462 - 533 Σf = 29 Larson/Farber 4th ed.

Determining the Midpoint Midpoint of a class Class Midpoint Frequency, f 30 - 101 65.5 5 102 - 173 137.5 3 174 - 245 209.5 246 - 317 281.5 7 318 - 389 353.5 4 390 - 461 425.5 1 462 - 533 497.5 Class width = 72

Determining the Relative Frequency Relative Frequency of a class Portion or percentage of the data that falls in a particular class. Larson/Farber 4th ed.

Determining the Relative Frequency continued Class Midpoint Frequency, f Relative Frequency 30 - 101 65.5 5 0.172 102 - 173 137.5 3 0.103 174 - 245 209.5 246 - 317 281.5 7 0.241 318 - 389 353.5 4 0.138 390 - 461 425.5 1 0.034 462 - 533 497.5 Σf = 29 1.000 Sum of Rel. Freq.

Determining the Cumulative Frequency Cumulative frequency of a class The sum of the frequency for that class and all previous classes. Class Midpoint Frequency, f Cumulative Frequency 30 - 101 65.5 5 102 - 173 137.5 3 8 174 - 245 209.5 13 246 - 317 281.5 7 20 318 - 389 353.5 4 24 390 - 461 425.5 1 25 462 - 533 497.5 29 Σf = Sum of class and previous class.

Expanded Frequency Distribution Class Midpoint Frequency, f Relative Frequency Cumulative 30 - 101 65.5 5 0.172 102 - 173 137.5 3 0.103 8 174 - 245 209.5 13 246 - 317 281.5 7 0.241 20 318 - 389 353.5 4 0.138 24 390 - 461 425.5 1 0.034 25 462 - 533 497.5 29 Σ =

Graphs of Frequency Distributions Frequency Histogram A bar graph that represents the frequency distribution. The horizontal scale is quantitative and measures the data values. The vertical scale measures the frequencies of the classes. Consecutive bars must touch. data values frequency Larson/Farber 4th ed.

Class Boundaries Class boundaries The numbers that separate classes without forming gaps between them. The distance from the upper limit of the first class to the lower limit of the second class is 102 – 101 = 1. Half this distance is 0.5. Class Boundaries 30 - 101 102 - 173 174 - 245 29.5 – 101.5 First class lower boundary = 30 – 0.5 = 29.5 First class upper boundary = 101 + 0.5 = 101.5 Larson/Farber 4th ed.

Class Boundaries Class Class boundaries Frequency, f 30 - 101 29.5 - 101.5 5 102 - 173 101.5 - 173.5 3 174 - 245 173.5 - 245.5 246 - 317 245.5 - 317.5 7 318 - 389 317.5 - 389.5 4 390 - 461 389.5 - 461.5 1 462 - 533 461.5 - 533.5 Larson/Farber 4th ed.

Example: Frequency Histogram Construct a frequency histogram for the book costs frequency distribution. Class Class boundaries Midpoint Frequency, f 30 - 101 29.5 - 101.5 65.5 5 102 - 173 101.5 - 173.5 137.5 3 174 - 245 173.5 - 245.5 209.5 246 - 317 245.5 - 317.5 281.5 7 318 - 389 317.5 - 389.5 353.5 4 390 - 461 389.5 - 461.5 425.5 1 462 - 533 461.5 - 533.5 497.5 Larson/Farber 4th ed. 21

Solution: Frequency Histogram (using Midpoints)

Graphs of Frequency Distributions Frequency Polygon A line graph that emphasizes the continuous change in frequencies. data values frequency Larson/Farber 4th ed.

Example: Frequency Polygon Construct a frequency polygon for the Books costs frequency distribution. Class Midpoint Frequency, f 30 - 101 65.5 5 102 - 173 137.5 3 174 - 245 209.5 246 - 317 281.5 7 318 - 389 353.5 4 390 - 461 425.5 1 462 - 533 497.5 Larson/Farber 4th ed.

Solution: Frequency Polygon The graph should begin and end on the horizontal axis, so extend the left side to one class width before the first class midpoint and extend the right side to one class width after the last class midpoint. This is bull! In most cases it is better to just show the data and not have false markers!

Graphs of Frequency Distributions Relative Frequency Histogram Has the same shape and the same horizontal scale as the corresponding frequency histogram. The vertical scale measures the relative frequencies, not frequencies. data values relative frequency Larson/Farber 4th ed.

Graphs of Frequency Distributions Cumulative Frequency Graph or Ogive A line graph that displays the cumulative frequency of each class at its upper class boundary. The upper boundaries are marked on the horizontal axis. The cumulative frequencies are marked on the vertical axis. data values cumulative frequency Larson/Farber 4th ed.

Constructing an Ogive Construct a frequency distribution that includes cumulative frequencies as one of the columns. Specify the horizontal and vertical scales. The horizontal scale consists of the upper class boundaries or upper limit. The vertical scale measures cumulative frequencies. Plot points that represent the upper class boundaries and their corresponding cumulative frequencies. Larson/Farber 4th ed.

Constructing an Ogive Connect the points in order from left to right. The graph should start at the lower boundary of the first class (cumulative frequency is zero) and should end at the upper boundary of the last class (cumulative frequency is equal to the sample size). Larson/Farber 4th ed.

Example: Ogive Construct an ogive for the book cost frequency distribution. Class Midpoint Frequency, f Cumulative Frequency 30 - 101 65.5 5 102 - 173 137.5 3 8 174 - 245 209.5 13 246 - 317 281.5 7 20 318 - 389 353.5 4 24 390 - 461 425.5 1 25 462 - 533 497.5 29

Solution: Ogive From the ogive, you can see that about 25 students spent $461 or less. The greatest increase in in cost occurs between $245 and $389. Larson/Farber 4th ed.

Section 2.1 Summary Constructed frequency distributions Constructed frequency histograms, frequency polygons, relative frequency histograms and ogives Larson/Farber 4th ed.