Frequency Distributions and Graphs Descriptive Statistics Frequency Distributions and Graphs

Introduction Data must be organized Present data Frequency distribution Present data Statistical graphs and charts

Organizing Data Frequency distribution Uses classes and frequency 3 types Categorical frequency distribution Nominal - or ordinal- type data Ungrouped frequency distribution Grouped frequency distribution

Construct a Categorical Distribution Make a table Class Tally Frequency Percent Tally the data Count the tallies – frequency Find the percentage % = f/n * 100 f – frequency n – total number

Example 2-1 Twenty-five army inductees were given a blood test to determine their blood type. Construct a frequency distribution for the data. AB B A O Step 1: Make a table Step 2: Tally the data Step 3: Count the tallies Step4: Find the percentage of values in each class

Example 2-1 Class A B O AB Tally IIIII IIII II IIII IIII IIII Frequency 5 7 9 4 Percent 5/25 * 100 =20 7/25*100=28 9/25*100=36 4/25*100=16

Ungrouped Frequency Distribution Range of data is relatively small Range = highest value – lowest value Single data values for each class Steps similar to categorical distribution Make a table Each unique data value is a class Tally - Frequency Percent or Relative Frequency Rel. freq. = # of ea. value/ total # of values

Example A survey was taken on Maple Avenue. In each of 20 homes, people were asked how many cars were registered to their households. The results were recorded as follows: 1, 2, 1, 0, 3, 4, 0, 1, 1, 1, 2, 2, 3, 2, 3, 2, 1, 4, 0, 0 Construct a frequency distribution for the data.

Solution Step 1: Make a table # of cars tally frequency rel. freq. Step 2: Tally – Frequency Step 3: Percent or Rel. Freq. # of Cars Tally Frequency Rel. Freq. IIII 4 4/20 = .20 1 IIII I 6 6/20 = .30 2 5 5/20 = .25 3 III 3/20 = .15 II 2/20 = .10

Grouped Frequency Distribution Range of data is large Classes are more than 1 unit in width Vocabulary Class width – difference between consecutive lower class limits or upper class limits 31 – 24 = 7 37 – 30 = 7 Class Limits 24 – 30 31 – 37 38 – 44 Lower class limits Upper class limits

Grouped Frequency Distribution Class boundaries Always end in a 5 1 decimal place to right of data If data are whole numbers - 0.5 from lower limit + 0.5 to upper limit Class limits should have the same # of dec. places as the data Class limits Class boundaries 24 – 30 23.5 – 30.5 31 – 37 30.5 – 37.5 38 – 44 37.5 – 44.5 -0.5 +0.5

Guidelines for Classes There should be between 5 and 20 classes Class width should be an odd number Midpoints will be integers instead of decimals Classes must be mutually exclusive No data value can fall into 2 classes Classes must be continuous No gaps Even if there is no data in a class it must be included Classes are exhaustive All data fits into a class Classes are equal in width Only exception open-ended class

Constructing a grouped freq. distribution Determine the number of classes Find the highest and lowest values Find the range Select the # of classes (5 – 20) Find the width w = range/# of classes Round up 3.2 becomes 4 3 becomes 4 Choose the lowest value as your starting point Add the class width to this lower limit Continue adding the width to the lower limits until you get the rest of the lower limits Find the first upper limit Subtract 1 from the second lower limit Continue adding the width to find the remaining upper limits

Example 2-2 - done in parts These data represent the record high temperatures for each of the 50 states. Construct a grouped frequency distribution for the data using 7 classes. 112 100 127 120 134 118 105 110 109 112 110 118 117 116 118 122 114 114 105 109 107 112 114 115 118 117 118 122 106 110 116 108 110 121 113 120 119 111 104 111 120 113 120 117 105 110 118 112 114 114

Class limits Determine the # of classes Find the highest and lowest values H = 134 L = 100 Find the range: H – L ; R = 134 – 100 = 34 Select # of classes (already asks for 7) Find the class width: w = R/7; w = 34/7 = 4.9 ~ 5 Choose a starting point – lowest value = 100 Add the class width to find next lower limit Continue adding the width until there are 7 Find the first upper limit – subtract 1 from the second lower limit 105 – 1 = 104 Add the width to the upper limits Class limits 100 – 104 105 – 109 110 – 114 115 – 119 120 – 124 125 – 129 130 - 134 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5

Class boundaries & Class midpoints Class limits Class boundaries Class midpoints 100 – 104 99.5 – 104.5 102 105 – 109 104.5 – 109.5 107 110 – 114 109.5 – 114.5 112 115 – 119 114.5 – 119.5 117 120 – 124 119.5 – 124.5 122 125 – 129 124.5 – 129.5 127 130 – 134 129.5 – 134.5 132 Class midpoints determined by adding the 2 class limits together then dividing by 2 Or Adding the 2 class boundaries together then dividing by 2 100 + 104 = 204/2 = 102 - .5 + .5

Constructing a grouped freq. distribution Tally the data Find the numerical frequencies from the tallies Find the cumulative frequencies Class limits bound Tally Freq Cum. Freq 100 – 104 99.5 – 104.5 II 2 105 – 109 104.5 – 109.5 IIII III 8 10 110 – 114 109.5 – 114.5 IIII IIII IIII III 18 28 115 – 119 114.5 – 119.5 IIII IIII III 13 41 120 – 124 119.5 – 124.5 IIII II 7 48 125 – 129 124.5 – 129.5 I 1 49 130 – 134 129.5 – 134.5 50 + + + + + +