This pdf includes the following topics:- Mean – Grouped Data Median and Interquartile Range – Grouped Data Quartiles Mode – Grouped Data Variance and Standard Deviation-Grouped Data
1. Lecture 2 – Grouped Data Calculation 1. Mean, Median and Mode 2. First Quantile, third Quantile and Interquantile Range.
2. Mean – Grouped Data Example: The following table gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company. Calculate the mean. Number f of order 10 – 12 4 13 – 15 12 16 – 18 20 19 – 21 14 n = 50 X is the midpoint of the Number f x fx class. It is adding the class of order limits and divide by 2. 10 – 12 4 11 44 13 – 15 12 14 168 x= ∑ fx = 8 3 2 = 1 6 .6 4 16 – 18 20 17 340 n 50 19 – 21 14 20 280 n = 50 = 832
3. Median and Interquartile Range – Grouped Data Step 1: Construct the cumulative frequency distribution. Step 2: Decide the class that contain the median. Class Median is the first class with the value of cumulative frequency equal at least n/2. Step 3: Find the median by using the following formula: ⎛ n ⎞ ⎜ 2 - F ⎟ M e d ia n = L m + ⎜ ⎟ i ⎜ f m ⎟ ⎝ ⎠ Where: n = the total frequency F = the cumulative frequency before class median f = the frequency of the class median m i = the class width Lm = the lower boundary of the class median
4. Example: Based on the grouped data below, find the median: Time to travel to work Frequency 1 – 10 8 11 – 20 14 21 – 30 12 31 – 40 9 41 – 50 7 1st Step: Construct the cumulative frequency distribution Time to travel Frequency Cumulative to work Frequency 1 – 10 8 8 11 – 20 14 22 21 – 30 12 34 31 – 40 9 43 41 – 50 7 50 n 50 = = 25 class median is the 3rd class 2 2 So, F = 22, fm = 12, L = 20.5 and i = 10 m
5. ⎛n ⎞ ⎜ - F ⎟ Median = Lm + ⎜ 2 ⎟i f ⎜ m ⎟ ⎝ ⎠ ⎛ 25 - 22 ⎞ = 21.5 + ⎜ ⎟ 10 ⎝ 12 ⎠ = 24 Thus, 25 persons take less than 24 minutes to travel to work and another 25 persons take more than 24 minutes to travel to work.
6. Using the same method of calculation as in the Median, we can get Q1 and Q3 equation as follows: ⎛n ⎞ ⎛ 3n ⎞ ⎜4-F ⎟ ⎜ 4 -F ⎟ Q1 = LQ1 + ⎜ ⎟i Q3 = LQ3 + ⎜ ⎟i ⎜ f Q1 ⎟ ⎜ f Q3 ⎟ ⎝ ⎠ ⎝ ⎠ Example: Based on the grouped data below, find the Interquartile Range Time to travel to work Frequency 1 – 10 8 11 – 20 14 21 – 30 12 31 – 40 9 41 – 50 7
7. 1st Step: Construct the cumulative frequency distribution Time to travel Frequency Cumulative to work Frequency 1 – 10 8 8 11 – 20 14 22 21 – 30 12 34 31 – 40 9 43 41 – 50 7 50 2nd Step: Determine the Q1 and Q3 n 50 ⎛n ⎞ Class Q 1 = = = 12 . 5 4 4 ⎜ 4 -F ⎟ Q1 = LQ1 + ⎜ ⎟i ⎜ fQ1 Class Q1 is the 2nd class ⎟ Therefore, ⎝ ⎠ ⎛ 12.5 - 8 ⎞ = 10.5 + ⎜ ⎟ 10 ⎝ 14 ⎠ = 13.7143
9. Mode – Grouped Data •Mode is the value that has the highest frequency in a data set. •For grouped data, class mode (or, modal class) is the class with the highest frequency. •To find mode for grouped data, use the following formula: ⎛ Δ1 ⎞ M o d e = Lmo + ⎜ ⎟i Δ ⎝ 1 + Δ 2 ⎠ i is the class width Δ1 is the difference between the frequency of class mode and the frequency of the class after the class mode Δ 2 is the difference between the frequency of class mode and the frequency of the class before the class mode Lmo is the lower boundary of class mode
10. Calculation of Grouped Data - Mode Example: Based on the grouped data below, find the mode Time to travel to work Frequency 1 – 10 8 11 – 20 14 21 – 30 12 31 – 40 9 41 – 50 7 Based on the table, Lmo = 10.5, Δ1 = (14 – 8) = 6, Δ 2 = (14 – 12) = 2 and i = 10 ⎛ 6 ⎞ M o d e = 1 0 .5 + ⎜ ⎟ 1 0 = 1 7 .5 ⎝ 6 + 2 ⎠
11. Mode can also be obtained from a histogram. Step 1: Identify the modal class and the bar representing it Step 2: Draw two cross lines as shown in the diagram. Step 3: Drop a perpendicular from the intersection of the two lines until it touch the horizontal axis. Step 4: Read the mode from the horizontal axis
12. Variance and Standard Deviation -Grouped Data ( ∑ fx ) 2 Population Variance: ∑ fx 2 − N σ2 = N ( ∑ fx ) 2 ∑ fx 2 − n Variance for sample data: s = 2 n −1 Standard Deviation: Population: σ2 = σ2 Sample: s2 = s2
13. Example: Find the variance and standard deviation for the following data: No. of order f 10 – 12 4 13 – 15 12 16 – 18 20 19 – 21 14 Total n = 50 Solution: No. of order f x fx fx2 10 – 12 4 11 44 484 13 – 15 12 14 168 2352 16 – 18 20 17 340 5780 19 – 21 14 20 280 5600 Total n = 50 832 14216
14. ( ∑ fx ) 2 ∑ fx 2 − n Variance, s2 = n −1 (832 ) 2 14216 − = 50 50 − 1 = 7.5820 Standard Deviation, s = s = 7.5820 = 2.75 2 Thus, the standard deviation of the number of orders received at the office of this mail-order company during the past 50 days is 2.75.
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