# Variation About the Mean Homework

Problem H1
For the allocation below, find and interpret the MAD, variance, and standard deviation.  You will need to find the mean first in order to calculate the deviation for each value in the set.

Problem H2
For the allocation below, find and interpret the MAD, variance, and standard deviation. (Note that in this problem the mean is not a whole number, so there will be a fractional deviation for each value. To simplify your work, round off each calculation to one decimal place.) What do these calculations tell you about the data in this problem as compared to the data in Problem H1? Kader, Gary (March, 1999). Means and MADS. Mathematics Teaching in the Middle School, 4 (6), 398-403.

Continued

Uccellini, John C. (November-December, 1996). Teaching the Mean Meaningfully. Mathematics Teaching in the Middle School, 2 (3), 112-115.

Teaching the Mean Meaningfully

Zawojewski, Judith and Shaughnessy, Michael (March, 2000). Mean and Median: Are They Really So Easy? Mathematics Teaching in the Middle School, 5(7), 436-440.

Mean and Median: Are They Really So Easy?
Continued

### Solution: Problem H1

First you’ll need to find the mean by adding all the values and dividing by 9:

6 + 6 + 9 + 6 + 6 + 7 + 8 + 9 + 6 = 63 / 9 = 7

The mean is 7. (Note that in this calculation, the stacks are not in order. You do not need to order this list before calculating the MAD, variance, or standard deviation.) Here is the table for calculating the MAD:  Number of Coins in Stack (x) Deviation from the Mean
(x – 7) Absolute Deviation from the Mean
|x – 7|  6 -1 1 6 -1 1 9 +2 2 6 -1 1 6 -1 1 7 0 0 8 +1 1 9 +2 2 6 -1 1 ______ ______ ______ 63 0 10 The MAD is 10 / 9, or approximately 1.11.

Here is the table for calculating the variance:  Number of Coins in Stack (x) Deviation from the Mean
(x – 7) Squared Deviation from the Mean
(x – 7)2  6 -1 1 6 -1 1 9 +2 4 6 -1 1 6 -1 1 7 0 0 8 +1 1 9 +2 4 6 -1 1 ______ ______ ______ 63 0 14 The variance is 14 / 9, or approximately 1.56.

The standard deviation is the square root of 1.56, which is approximately 1.25. Note that the MAD and the standard deviation are roughly the same.

### Solution: Problem H2

First you’ll need to find the mean by adding all the values and dividing by 9:

1 + 4 + 10 + 4 + 4 + 6 + 7 + 10 + 4 = 50 / 9 = 5.6 (to one decimal place)

We will use 5.6 for the mean. Here is the table for calculating the MAD:  Number of Coins in Stack (x) Deviation from the Mean
(x – 5.6) Absolute Deviation from the Mean
|x – 5.6|  1 -4.6 4.6 4 -1.6 1.6 10 +4.4 4.4 4 -1.6 1.6 4 -1.6 1.6 6 +0.4 0.4 7 +1.4 1.4 10 +4.4 4.4 4 -1.6 1.6 ______ ______ ______ 50 0 21.6 The MAD is approximately 21.6 / 9 = 2.4 (to one decimal place). (If you are using fractions, the exact answer is 194/81.)

Here is the table for calculating the variance:  Number of Coins in Stack (x) Deviation from the Mean
(x – 5.6) Squared Deviation from the Mean
|x – 5.6|2  1 -4.6 21.16 4 -1.6 2.56 10 +4.4 19.36 4 -1.6 2.56 4 -1.6 2.56 6 +0.4 0.16 7 +1.4 1.96 10 +4.4 19.36 4 -1.6 2.56 ______ ______ ______ 50 0 72.24 The variance is 72.24 / 9 = 8.0 (to one decimal place). (If you were using fractions, the exact answer is 650/81.)

The standard deviation is the square root of 8, which is 2.8 (to one decimal place). (The exact fractional answer, the square root of 650/81, cannot be expressed by a rational number, since 650 is not a square number.)

The results of Problems H1 and H2 suggest that the data in Problem H2 have greater variation about the mean than the data in Problem H1.