Show Explained with Three Examples The Race and the Naughty PuppyThis starts with some raw data (not a grouped frequency yet) ...
Alex timed 21 people in the sprint race, to the nearest second:59, 65, 61, 62, 53, 55, 60, 70, 64, 56, 58, 58, 62, 62, 68, 65, 56, 59, 68, 61, 67 To find the Mean Alex adds up all the numbers, then divides by how many numbers: Mean = 59 + 65 + 61 + 62 + 53 + 55 + 60 + 70 + 64 + 56 + 58 + 58 + 62 + 62 + 68 + 65 + 56 + 59 + 68 + 61 + 6721 To find the Median Alex places the numbers in value order and finds the middle number. In this case the median is the 11th number: 53, 55, 56, 56, 58, 58, 59, 59, 60, 61, 61, 62, 62, 62, 64, 65, 65, 67, 68, 68, 70 Median = 61 To find the Mode, or modal value, Alex places the numbers in value order then counts how many of each number. The Mode is the number which appears most often (there can be more than one mode): 53, 55, 56, 56, 58, 58, 59, 59, 60, 61, 61, 62, 62, 62, 64, 65, 65, 67, 68, 68, 70 62 appears three times, more often than the other values, so Mode = 62 Grouped Frequency TableAlex then makes a Grouped Frequency Table:
So 2 runners took between 51 and 55 seconds, 7 took between 56 and 60 seconds, etc Oh No!
Suddenly all the original data gets lost (naughty pup!) ... can we help Alex calculate the Mean, Median and Mode from just that table? The answer is ... no we can't. Not accurately anyway. But, we can make estimates. Estimating the Mean from Grouped DataSo all we have left is:
The groups (51-55, 56-60, etc), also called class intervals, are of width 5 The midpoints are in the middle of each class: 53, 58, 63 and 68 We can estimate the Mean by using the midpoints.
So, how does this work? Think about the 7 runners in the group 56 - 60: all we know is that they ran somewhere between 56 and 60 seconds:
So we take an average and assume that all seven of them took 58 seconds. Let's now make the table using midpoints:
Our thinking is: "2 people took 53 sec, 7 people took 58 sec, 8 people took 63 sec and 4 took 68 sec". In other words we imagine the data looks like this: 53, 53, 58, 58, 58, 58, 58, 58, 58, 63, 63, 63, 63, 63, 63, 63, 63, 68, 68, 68, 68 Then we add them all up and divide by 21. The quick way to do it is to multiply each midpoint by each frequency:
And then our estimate of the mean time to complete the race is: Estimated Mean = 128821 = 61.333... Very close to the exact answer we got earlier. Estimating the Median from Grouped DataLet's look at our data again:
The median is the middle value, which in our case is the 11th one, which is in the 61 - 65 group: We can say "the median group is 61 - 65" But if we want an estimated Median value we need to look more closely at the 61 - 65 group.
We call it "61 - 65", but it really includes values from 60.5 up to (but not including) 65.5. Why? Well, the values are in whole seconds, so a real time of 60.5 is measured as 61. Likewise 65.4 is measured as 65. At 60.5 we already have 9 runners, and by the next boundary at 65.5 we have 17 runners. By drawing a straight line in between we can pick out where the median frequency of n/2 runners is: And this handy formula does the calculation: Estimated Median = L + (n/2) − BG × w where:
For our example:
Estimated Median= 60.5 + (21/2) − 9 8 × 5 = 60.5 + 0.9375 = 61.4375 Estimating the Mode from Grouped DataAgain, looking at our data:
We can easily find the modal group (the group with the highest frequency), which is 61 - 65 We can say "the modal group is 61 - 65" But the actual Mode may not even be in that group! Or there may be more than one mode. Without the raw data we don't really know. But, we can estimate the Mode using the following formula: Estimated Mode = L + fm − fm-1(fm − fm-1) + (fm − fm+1) × w where:
In this example:
Estimated Mode= 60.5 + 8 − 7(8 − 7) + (8 − 4) × 5 = 60.5 + (1/5) × 5 = 61.5
Our final result is:
(Compare that with the true Mean, Median and Mode of 61.38..., 61 and 62 that we got at the very start.) And that is how it is done. Now let us look at two more examples, and get some more practice along the way! Baby Carrots ExampleExample: You grew fifty baby carrots using special soil. You dig them up and measure their lengths (to the nearest mm) and group the results:
Mean
Estimated Mean = 853050 = 170.6 mm MedianThe Median is the mean of the 25th and the 26th length, so is in the 170 - 174 group:
Estimated Median= 169.5 + (50/2) − 219 × 5 = 169.5 + 2.22... = 171.7 mm (to 1 decimal) ModeThe Modal group is the one with the highest frequency, which is 175 - 179:
Estimated Mode= 174.5 + 11 − 9(11 − 9) + (11 − 6) × 5 = 174.5 + 1.42... = 175.9 mm (to 1 decimal) Age ExampleAge is a special case. When we say "Sarah is 17" she stays "17" up until her eighteenth birthday. This changes the midpoints and class boundaries. Example: The ages of the 112 people who live on a tropical island are grouped as follows:
A child in the first group 0 - 9 could be almost 10 years old. So the midpoint for this group is 5 not 4.5 The midpoints are 5, 15, 25, 35, 45, 55, 65, 75 and 85 Similarly, in the calculations of Median and Mode, we will use the class boundaries 0, 10, 20 etc Mean
Estimated Mean = 3360112 = 30 MedianThe Median is the mean of the ages of the 56th and the 57th people, so is in the 20 - 29 group:
Estimated Median= 20 + (112/2) − 4123 × 10 = 20 + 6.52... = 26.5 (to 1 decimal) ModeThe Modal group is the one with the highest frequency, which is 20 - 29:
Estimated Mode= 20 + 23 − 21(23 − 21) + (23 − 16) × 10 = 20 + 2.22... = 22.2 (to 1 decimal) Summary
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