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Descriptive Statistics

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

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

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

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

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

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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

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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.

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Step 1: Make a table # of cars tally frequency rel. freq. Step 2: Tally – Frequency Step 3: Percent or Rel. Freq. # of CarsTallyFrequencyRel. Freq. 0IIII44/20 =.20 1IIII I66/20 =.30 2IIII55/20 =.25 3III33/20 =.15 4II22/20 =.10 Solution

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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

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Grouped Frequency Distribution Class limitsClass boundaries 24 – 3023.5 – 30.5 31 – 3730.5 – 37.5 38 – 4437.5 – 44.5 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 -0.5 +0.5

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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

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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

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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

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Class limits 100 – 104 105 – 109 110 – 114 115 – 119 120 – 124 125 – 129 130 - 134 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 + 5

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Class boundaries & Class midpoints Class limits Class boundaries Class midpoints 100 – 10499.5 – 104.5102 105 – 109104.5 – 109.5107 110 – 114109.5 – 114.5112 115 – 119114.5 – 119.5117 120 – 124119.5 – 124.5122 125 – 129124.5 – 129.5127 130 – 134129.5 – 134.5132 -.5 +.5 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

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Constructing a grouped freq. distribution Tally the data Find the numerical frequencies from the tallies Find the cumulative frequencies Class limits Class bound TallyFreq Cum. Freq 100 – 104 99.5 – 104.5 II22 105 – 109 104.5 – 109.5 IIII III810 110 – 114 109.5 – 114.5 IIII IIII IIII III 1828 115 – 119 114.5 – 119.5 IIII IIII III1341 120 – 124 119.5 – 124.5 IIII II748 125 – 129 124.5 – 129.5 I149 130 – 134 129.5 – 134.5 I150 + + + + + +

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