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Mean Median Mode
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# Grouped Mean Median Mode

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Mean Median Mode for Grouped Data containing Class Intervals and Bins in Statistics

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### Grouped Mean Median Mode

1. 1. Image Source: http://www.blogspot.com 0 - 3 2 4 - 7 3 8 – 11 8 12 - 15 3 16 - 19 2 Number of Cappuccinos Freq Image Source: http://www.espressospot.com
2. 2. 0 - 3 2 4 - 7 3 8 – 11 8 12 - 15 3 16 - 19 2 Number of Cappuccinos Freq Image Source: http://www.wordpress.com A survey was conducted at a Cafe which sells food and coffees. The Barista suggested that they get a bigger machine to cope with high demand. A bigger machine is very expensive to buy, and so the owner had a two day survey done to find out how many Cappuccinos were being made per hour in the Cafe. From the survey results, they would be able to do some Statistics, and better understand the current problem situation.
3. 3. The following is a Histogram of the Number of Coffees made for each hour that the coffee shop was open over two days. We want to find the Average Number of Coffees per hour. Image Source: http://www.blogspot.com 0 - 3 2 4 - 7 3 8 – 11 8 12 - 15 3 16 - 19 2 Number of Cappuccinos Freq 0–3 8 6 4 2 0 Number of Cappuccinos Made Per Hour
4. 4. The “Modal Class” is the group which occurs the most times. It is the biggest frequency group and forms tallest column. Image Source: http://www.blogspot.com 0 - 3 2 4 - 7 3 8 – 11 8 12 - 15 3 16 - 19 2 Number of Cappuccinos Freq 0–3 8 6 4 2 0 Number of Cappuccinos Made Per Hour The “Modal Class” is the group 8 to 11 Cappuccinos.
5. 5. There can be several groups in the “Modal Class” if we have the biggest frequency value occur multiple times. Image Source: http://metrio.com.au 0 - 3 2 4 - 7 3 8 – 11 3 12 - 15 3 16 - 19 2 Number of Café Lattes Freq 0–3 8 6 4 2 0 Number of Café Lattes Made Per Hour “Modal Class” is 4-7, 8-11, and 12-15 Café Lattes.
6. 6. The Median is halfway through our Groups of Interval Data. Cappuccinos f 0-3 2 4-7 3 8-11 8 12-15 3 16-19 2 TOTAL 18 We find the Median by finding our Half-Way Position within the Frequencies, which is NOT 18 / 2 = 9  1,2,3,4,5,6,7,8,9, 10,11,12,13,14,15,16,17,18 The middle of our Frequencies is at the 9.5th value. 9.5
7. 7. The Median is halfway through our Groups of Interval Data. Cappuccinos f 0-3 2 4-7 3 8-11 8 12-15 3 16-19 2 TOTAL 18 We can find our Half-Way Position using the formula: (Total Frequency + 1) / 2 = (18 + 1) / 2 = 9.5 1,2,3,4,5,6,7,8,9, 10,11,12,13,14,15,16,17,18 The middle of our Frequencies is at the 9.5th value. 9.5
8. 8. To find the Median, we also need to do Cumulative Addition. Cappuccinos f Cumulative f 0-3 2 4-7 3 8-11 8 12-15 3 16-19 2 TOTAL 18 We add a new column where we do continual Totalling of the “f” Frequency values. We add on an extra column, and add the frequencies. But first we need to learn how to do Cumulative Addition.
9. 9. Our friend Glen is busking and getting money for each song. Song \$ Cumulative \$ Let It Be 15 15 Get Back 4 15 + 4 19 Lady Madonna 10 15 + 4 + 10 29 Imagine 20 15+4+10+20 49 TOTAL 49 After he has finished each song, we can see how many total dollars he has. The last values must be identical (eg. both 49), or else we have made an error.
10. 10. A quicker way to add up is using the “Zig Zag” pattern. Song \$ Cumulative \$ Let It Be 15 15 Get Back 4 19 Lady Madonna 10 29 Imagine 20 49 TOTAL 49 After he has finished each song, we can see how many \$ he has earned so far. The last values must be identical (eg. both 49), or else we have made an error. + + +
11. 11. 1st step is add the extra column, and do Cumulative Addition. Cappuccinos f Cumulative f 0-3 2 2 4-7 3 5 8-11 8 13 12-15 3 16 16-19 2 18 TOTAL 18 We add a new column where we do continual Totalling of the “f” Frequency values. (Use Zig Zag Pattern) The other step is to find the half-way frequency using ( Total f + 1 ) / 2 = (18 + 1 ) / 2 = 9.5th position.
12. 12. Half-Way through the Frequency is (18 + 1) /2 = 9.5 Cappuccinos f Cumulative f 0-3 2 2 4-7 3 5 8-11 8 13 12-15 3 16 16-19 2 18 TOTAL 18 The 9.5th position occurs during the 8-11 Interval. In this Interval we pass through 9.5 on the way to reaching 13. The Median Class is 8-11
13. 13. We find the Average Number of Coffees per Hour by adding two new columns to our Frequency Table and using a Formula. Cappuccinos Freq Interval Midpoint Freq x Midpt 0-3 2 4-7 3 8-11 8 12-15 3 16-19 2 TOTALS 18 We find all of our Interval Midpoints first, and then we Multiply each Frequency x Midpoint value as our second step
14. 14. We find the Average Number of Coffees per Hour by adding two new columns to our Frequency Table and using a Formula. Cappuccinos Freq Interval Midpoint 0-3 2 0,1, 2,3 -> 1.5 4-7 3 4,5, 6,7 -> 5.5 8-11 8 8,9, 10,11 -> 9.5 12-15 3 12,13, 14,15 13.5 16-19 2 16,17, 18,19 17.5 TOTALS 18 We find Interval Midpoints by getting the exact middle (or Median) of each of our data classes, or data groups. Eg. Exactly half way between 0 to 3 is the value of 1.5
15. 15. We can also find the Midpoints by adding the Lower end to the Upper end of an Interval, and then dividing by 2. Cappuccinos Freq Interval Midpoint 0-3 2 (0 + 3) /2 -> 1.5 4-7 3 (4 + 7) /2 -> 5.5 8-11 8 (8+11) /2 -> 9.5 12-15 3 (12+15) /2 -> 13.5 16-19 2 (16+19) /2 -> 17.5 TOTALS 18 We find Interval Midpoints by getting the exact middle (or Median) of each of our data classes, or data groups. Eg. Exactly half way between 0 to 3 is the value of 1.5
16. 16. We complete the final column by Multiplying the Frequency by the Midpoint Value for each row of our class groups. Cappuccinos Freq Interval Midpoint Freq x Midpt 0-3 2 0,1,2,3 -> 1.5 2 x 1.5 = 3 4-7 3 4,5,6,7 -> 5.5 3 x 5.5 = 16.5 8-11 8 8,9,10,11 -> 9.5 8 x 9.5 = 76 12-15 3 12,13,14,15 13.5 3 x 13.5 = 40.5 16-19 2 16,17,18,19 17.5 2 x 17.5 = 35 TOTALS 18 171 The 3rd and final step is to use the Grouped Mean Formula: MEAN Average = Total of (Freq x Midpt) / Total Frequency
17. 17. We find the Average Number of Coffees per Hour by adding two new columns to our Frequency Table and using a Formula. Cappuccinos Freq Interval Midpoint Freq x Midpt 0-3 2 1.5 2 x 1.5 = 3 4-7 3 5.5 3 x 5.5 = 16.5 8-11 8 9.5 8 x 9.5 = 76 12-15 3 13.5 3 x 13.5 = 40.5 16-19 2 17.5 2 x 17.5 = 35 TOTALS 18 171 MEAN Average = Total of (Freq x Midpt) / Total Frequency = 171 / 18 = 10 cappuccinos per hour
18. 18. We use some specific Mathematical Notation as follows: Groups like 0-3 are called “Intervals” eg. 0 up to 3 = 0,1,2,3 f = frequency, or how many we have of a particular interval x = midpoint, which is (Low + High) /2 for each interval is the sum of the frequencies (eg. Total Frequency) is the Total of all the Frequency x Midpoint values MEAN Average for Grouped Data = f fx f fx
19. 19. Using Standard Statistics Notation, we obtain the following: Cappuccinos f x f x 0-3 2 1.5 2 x 1.5 = 3 4-7 3 5.5 3 x 5.5 = 16.5 8-11 8 9.5 8 x 9.5 = 76 12-15 3 13.5 3 x 13.5 = 40.5 16-19 2 17.5 2 x 17.5 = 35 TOTALS = 18 = 171 MEAN = = 171 / 18 = 10 cappuccinos per hour f fx f fx
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