Mean, Median, & Mode:
The mean, median, and mode are called measures of central tendency.
Mode – value or property that occurs most frequently in the data.
- not the most stable way of looking at the data.
Ex. 4, 6, 6, 7, 8, 9
The mode would be 6 since it occurs most frequently.
*** If an equal frequency occurs then there is no mode.
Median – central value of an ordered distribution.
- obtained by ordering the data from smallest to largest, then finding the middle
value.
- More stable then the mode, but does not indicate the range of values above or
below it. (Example p. 112)
*** If there are an odd number of values in your distribution, the central value is the
median.
Ex. 12, 13, 16, 17, 19, 22, 35, 44, 59
The median is 19.
*** If there is an even number of values in your distribution, obtain the median by taking
the average of the two central values.
Ex. 27, 35, 44, 56, 67, 78, 89
The median is 50.
Mean – average that uses the exact value of each entry.
- can be affected by extreme values.
- add the values of all the entries and then divide by the number of entries.
Ex. 113, 116, 125, 135, 110, 109, 100
The mean is 115.4286 (rounded).
For each of the following, calculate the mode, median, and mean. Round to the nearest
tenth if necessary!
1) 44, 78, 91, 111, 86, 52, 57, 67, 108, 138, 11, 67, 92, 88, 75, 82, 79, 106, 111, 111,
134, 222, 45, 74, 111, 67, 92, & 45.
2) 567, 671, 670, 733, 563, 563, 672, 777, 782, 645, 375, 226, 973, 567, 711, 896,
678, 722, 917, 888, 777, 666, 335, 762, & 937.
3) 256, 102, 673, 834, 883, 991, 202, 907, 563, 444, 167, 783, 927, 863, 723, 829,
283, 923, 829, 839, 903, 920, 526, 673, 738, 672, 721, 452, & 233.
4) 1024, 1089, 8923, 6745, 6723, 7745, 8930, 4567, 8934, 8374, 8738, 8397, 7378,
9374, 7837, 8922, 2222, 7838, 7836, 7384, 7384, 7384, 7238, 2893, 2678, 2893,
8298, 7872, 2737, 1102, & 2233.
5) 4444, 2891, 1727, 2828, 4949, 2780, 2834, 4544, 4564. 1371, 7383, 2767, 2893,
8928, 2839, 7278, 2829, 9239, 8289, 2892, 8928, 1561 & 3330.
*** When we compute the mean, we add up or sum all of our entries. There is a notation
to indicate sum, Σ (sigma, the Greek letter S).
If we use x to represent any value in the data set, we can now use a formula for sample
mean.
Sample Mean = x bar = Σx
n
where n represents the total number of entries.
Try this example:
Each month during a period of 2 years, Mel traveled the following amount of
miles to work:
24
45
23
66
42
45
23
36
33
31
30
31
43
22
23
34
32
45
41
40
42
26
53
11
a) What is the value of n?
b) What is the value of Σx?
c) Compute the mean (x bar).
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The calculator can be a very valuable tool when we are working with measures of central
tendency. This example will show you how quickly and easily our calculations can be
made.
An accountant is trying to become a partner in the firm in which he works. He has been
told by his supervisor to calculate the number of hours that he has worked monthly over
the course of the year. He looks at his time sheets and comes up with the following data:
88.5 66.5 100
75
88.5 96.6 77
88
92.5 96
55.5 100
Calculate the sample mean, median, and mode for the data above.
Solution:
First, we need to get all of the entries into the calculator. We can do this by pressing the
STAT button. Now we select #1 EDIT and we should view a chart with our cursor under
L1. We will now enter our entries under L1.
Now that we have all of the data entered, we will once again press STAT and scroll over
to CALC. We will select #1 1-VAR STATS, which brings us back to the main screen
with that heading. Press ENTER to view the calculation for the data.
Notice, the calculator has computed x bar and Σx, as well as other tools which we will
use later in the course. X bar is our sample mean and Σx is the sum of all of the entries.
We still need to calculate the median. To help us obtain the median we obviously need to
put the data into order. The calculator does this for us. Press STAT and select #2 SortA
(this tells the calculator to sort the data in ascending order). Again we are back at the
main screen. We now have to enter the list number that we would like in ascending
order. Press 2ND STAT and select #1 L1. Then press ENTER at the main screen and go
back to your list and you will notice the data is now ordered for you.
Unfortunately, you will have to calculate the mode yourself.
Sample Mean = ________
Median = _________
Mode = _________
For each of the following calculate the sample mean, median, and mode. Round 4
decimal places if necessary.:
1) 45, 67, 56, 56.8, 56.3, 55, 45, 67, 82, 36, 46, 75, 75.4, 88.9, 45, 33, 67, 64, 77.3,
& 66.
2) 53, 67, 72.3, 88.8, 72, 73, 74.5, 66.3, 78, 88, 67.3, 67.2, 56.2, 67, 33, 56, & 56.
3) 67, 74, 77, 88, 93, 82, 73, 64, 75, 88, 36.5, 67.4, 77.3, 67.4, 67.3, 66, 22, 45, 99,
& 100.
4) 104, 106, 111, 123, 101, 126, 177, 134, 156, 167, 190, 174, 166, 166, 172, 188 &
156.
5) 333, 333, 333, 356, 378, 399, 399, 367, 366, 383, 371, 378, 399, & 344.
6) 245.6, 256.9, 278.9, 266, 288, 298.1, 278.7, 278.7, 288.9, 300, 287.6, 286.5,
288.9, & 202.5.
7) 901, 977, 976, 999.9, 928.6, 923, 945.9, 992.7, 901, 902, 903, 907.8, 977, & 945.
8) 777, 888, 999, 222, 333, 444, 555, 666, 666.5, 777.5, 888.5, 999.5, 222.5, 111.5,
111, 333.5, 444.5, & 555.5.
9) 564, 123, 789, 734.6, 22.3, 11.9, 78.4, 56.7, 56.78, 55.34, 88.97, 323.72, 88.96, &
44.95.
10) .67, .993, .9256, .9562, .9923, .0673, .8823, .5523, .827, .7736, .8234, .7723,
.76283, & .5555.
Resistant Measure – measure that is not influenced by extremely high or low data values.
The mean is not a resistant measure of central tendency because we can make the mean
as large as we want by increasing the size of only one data value. The median is a more
resistant measure, but it is not sensitive to the specific size of a data value.