Probability and Statistics: Grouped Frequency Distribution

Distribución de frecuencias de datos agrupados

1. ING. GILBERTO ANUAR GARCÍA GARCÍA PROBABILITY AND STATISTICS “FREQUENCY DISTRIBUTION”

2. U1 FREQUENCY DISTRIBUTION 2 DATA SORTING It’s the process that involves arranging the data into some meaningful order to make it easier to understand, analyze or visualize. QUALITATIVE VARIABLES QUANTITATIVE VARIABLES Increasing (smallest to largest value) Decreasing (higher to lower value) Alphabetical A – Z Alphabetical Z – A From the most to the least repeated value From least to most repeated value

3. U1 FREQUENCY DISTRIBUTION 3 FREQUENCY DISTRIBUTION: UNGROUPED VS. GROUPED When the sample size (n) is finite and the number of different data is small (n≤ 10), it is easy to do an analysis of the data by taking each of the different data and ordering them qualitatively or quantitatively. UNGROUPED FREQUENCY DISTRIBUTION When the sample size is considerable or large and the numerical data are very diverse ( n>15 ), it is convenient to group the data in such a way that it allows to establish patterns, trends or regularities of the observed values. Class Intervals: These are the intervals in which the observed values are grouped and ordered. Each of these intervals is delimited (bounded) by two extreme values that we call limits. GROUPED FREQUENCY DISTRIBUTION

4. U1 GROUPED FREQUENCY DISTRIBUTION 4 STEPS TO BUILD FREQUENCY INTERVALS 1 . Determine the number of bins   Sturge’s rule Velleman formula       [recommended for small sample sizes ( n<50)] 2. Calculate the range     3. Calculate the class width of each interval    

5. U1 GROUPED FREQUENCY DISTRIBUTION 5 STEPS TO BUILD FREQUENCY INTERVALS 4. Construct the class intervals Class intervals are numeric sets and must be exclusive and exhaustive; that is, if a data belongs to a certain interval, it cannot longer belong to another, this means exclusive and also each and every one of the data must be contained in one of the intervals, this gives them the value of exhaustive. 5. Calculate the class mark or mid-point    

6. U1 EXAMPLE A group of researchers from the public security took a random sample of the speeds ( km/h ) registered by 30 vehicles on the I-45 highway, in order to establish new maximum speed limits. The following data shows the results obtained: 90 99 104 99 119 98 95 112 95 120 100 90 116 96 114 108 98 118 100 106 114 100 112 106 100 115 111 105 114 97 6 Construct a grouped frequency distribution table. Solution: (1)   (2)   (3)  

7. U1 EXAMPLE 7 Solution: CLASS INTERVALS ABSOLUTE FREQUENCY CUMULATIVE FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY CLASS MARK [90 – 95) 2 2 0.07 0.07 92.5 [95 – 100) 8 10 0.27 0.34 97.5 [100 – 105) 5 15 0.17 0.51 102.5 [105 – 110) 4 19 0.13 0.64 107.5 [110 – 115) 6 25 0.20 0.84 112.5 [115 – 120) 5 30 0.16 1.00 117.5 CLASS INTERVALS [90 – 95) 2 2 0.07 0.07 92.5 [95 – 100) 8 10 0.27 0.34 97.5 [100 – 105) 5 15 0.17 0.51 102.5 [105 – 110) 4 19 0.13 0.64 107.5 [110 – 115) 6 25 0.20 0.84 112.5 [115 – 120) 5 30 0.16 1.00 117.5 (4) 90 99 104 99 119 98 95 112 95 120 100 90 116 96 114 108 98 118 100 106 114 100 112 106 100 115 111 105 114 97 (5)