Question

find the mad of the data set.
99, 90, 88, 95, 86, 94
10, 7, 5, 14, 19, 32, 25
4, 4, 2, 1, 2, 4, 2, 5
17, 15, 12, 12, 14
16, 12, 20, 15, 20, 11, 25
6, 3, 2, 4, 8, 11, 9, 13

Explanation:

Step1: Find the mean of the data set 99, 90, 88, 95, 86, 94

First, sum the data: \(99 + 90 + 88 + 95 + 86 + 94 = 552\)
Then, divide by the number of data points (\(n = 6\)): \(\text{Mean} = \frac{552}{6} = 92\)

Step2: Find the absolute deviations from the mean

\(\vert 99 - 92 \vert = 7\), \(\vert 90 - 92 \vert = 2\), \(\vert 88 - 92 \vert = 4\), \(\vert 95 - 92 \vert = 3\), \(\vert 86 - 92 \vert = 6\), \(\vert 94 - 92 \vert = 2\)

Step3: Find the mean of these absolute deviations (MAD)

Sum the absolute deviations: \(7 + 2 + 4 + 3 + 6 + 2 = 24\)
Divide by \(n = 6\): \(\text{MAD} = \frac{24}{6} = 4\)

(For other data sets, follow the same steps: 1. Calculate mean, 2. Find absolute deviations from mean, 3. Calculate mean of absolute deviations)

For data set 10, 7, 5, 14, 19, 32, 25:

Step1: Mean

Sum: \(10 + 7 + 5 + 14 + 19 + 32 + 25 = 112\), \(n = 7\), Mean \(= \frac{112}{7} = 16\)

Step2: Absolute deviations

\(\vert 10 - 16 \vert = 6\), \(\vert 7 - 16 \vert = 9\), \(\vert 5 - 16 \vert = 11\), \(\vert 14 - 16 \vert = 2\), \(\vert 19 - 16 \vert = 3\), \(\vert 32 - 16 \vert = 16\), \(\vert 25 - 16 \vert = 9\)

Step3: MAD

Sum: \(6 + 9 + 11 + 2 + 3 + 16 + 9 = 56\), MAD \(= \frac{56}{7} = 8\)

For data set 4, 4, 2, 1, 2, 4, 2, 5:

Step1: Mean

Sum: \(4 + 4 + 2 + 1 + 2 + 4 + 2 + 5 = 24\), \(n = 8\), Mean \(= \frac{24}{8} = 3\)

Step2: Absolute deviations

\(\vert 4 - 3 \vert = 1\) (three times), \(\vert 2 - 3 \vert = 1\) (three times), \(\vert 1 - 3 \vert = 2\), \(\vert 5 - 3 \vert = 2\)

Step3: MAD

Sum: \(1\times3 + 1\times3 + 2 + 2 = 3 + 3 + 2 + 2 = 10\), MAD \(= \frac{10}{8} = 1.25\)

For data set 17, 15, 12, 12, 14:

Step1: Mean

Sum: \(17 + 15 + 12 + 12 + 14 = 70\), \(n = 5\), Mean \(= \frac{70}{5} = 14\)

Step2: Absolute deviations

\(\vert 17 - 14 \vert = 3\), \(\vert 15 - 14 \vert = 1\), \(\vert 12 - 14 \vert = 2\) (two times), \(\vert 14 - 14 \vert = 0\)

Step3: MAD

Sum: \(3 + 1 + 2 + 2 + 0 = 8\), MAD \(= \frac{8}{5} = 1.6\)

For data set 16, 12, 20, 15, 20, 11, 25:

Step1: Mean

Sum: \(16 + 12 + 20 + 15 + 20 + 11 + 25 = 119\), \(n = 7\), Mean \(= \frac{119}{7} = 17\)

Step2: Absolute deviations

\(\vert 16 - 17 \vert = 1\), \(\vert 12 - 17 \vert = 5\), \(\vert 20 - 17 \vert = 3\) (two times), \(\vert 15 - 17 \vert = 2\), \(\vert 11 - 17 \vert = 6\), \(\vert 25 - 17 \vert = 8\)

Step3: MAD

Sum: \(1 + 5 + 3 + 3 + 2 + 6 + 8 = 28\), MAD \(= \frac{28}{7} = 4\)

For data set 6, 3, 2, 4, 8, 11, 9, 13:

Step1: Mean

Sum: \(6 + 3 + 2 + 4 + 8 + 11 + 9 + 13 = 56\), \(n = 8\), Mean \(= \frac{56}{8} = 7\)

Step2: Absolute deviations

\(\vert 6 - 7 \vert = 1\), \(\vert 3 - 7 \vert = 4\), \(\vert 2 - 7 \vert = 5\), \(\vert 4 - 7 \vert = 3\), \(\vert 8 - 7 \vert = 1\), \(\vert 11 - 7 \vert = 4\), \(\vert 9 - 7 \vert = 2\), \(\vert 13 - 7 \vert = 6\)

Step3: MAD

Sum: \(1 + 4 + 5 + 3 + 1 + 4 + 2 + 6 = 26\), MAD \(= \frac{26}{8} = 3.25\)

Answer:

• For 99, 90, 88, 95, 86, 94: \(4\)
• For 10, 7, 5, 14, 19, 32, 25: \(8\)
• For 4, 4, 2, 1, 2, 4, 2, 5: \(1.25\)
• For 17, 15, 12, 12, 14: \(1.6\)
• For 16, 12, 20, 15, 20, 11, 25: \(4\)
• For 6, 3, 2, 4, 8, 11, 9, 13: \(3.25\)