Question

calculating mean absolute deviation
the hours students reported practicing their instruments last grading quarter were 29, 20, 36, 44, 30, 32, and 40.
the mean hours students reported practicing their instruments is
the mean absolute deviation for the data is
0
6
7
33

Explanation:

Part 1: Calculating the Mean

Step 1: Sum the data values

The data values are 29, 20, 36, 44, 30, 32, and 40. We sum them up:
\(29 + 20 + 36 + 44 + 30 + 32 + 40\)
First, \(29 + 20 = 49\), \(49 + 36 = 85\), \(85 + 44 = 129\), \(129 + 30 = 159\), \(159 + 32 = 191\), \(191 + 40 = 231\). So the sum is \(231\).

Step 2: Divide by the number of data points

There are 7 data points. The mean \(\bar{x}\) is given by \(\bar{x}=\frac{\text{Sum of data values}}{\text{Number of data points}}\).
So \(\bar{x}=\frac{231}{7}=33\).

Part 2: Calculating the Mean Absolute Deviation (MAD)

Step 1: Find the absolute deviation for each data point

The mean \(\bar{x} = 33\). For each data point \(x_i\), calculate \(|x_i - \bar{x}|\):

• For \(29\): \(|29 - 33| = |-4| = 4\)
• For \(20\): \(|20 - 33| = |-13| = 13\)
• For \(36\): \(|36 - 33| = |3| = 3\)
• For \(44\): \(|44 - 33| = |11| = 11\)
• For \(30\): \(|30 - 33| = |-3| = 3\)
• For \(32\): \(|32 - 33| = |-1| = 1\)
• For \(40\): \(|40 - 33| = |7| = 7\)

Step 2: Sum the absolute deviations

Now we sum these absolute deviations: \(4 + 13 + 3 + 11 + 3 + 1 + 7\)
First, \(4 + 13 = 17\), \(17 + 3 = 20\), \(20 + 11 = 31\), \(31 + 3 = 34\), \(34 + 1 = 35\), \(35 + 7 = 42\). So the sum of absolute deviations is \(42\).

Step 3: Divide by the number of data points

The mean absolute deviation (MAD) is \(\frac{\text{Sum of absolute deviations}}{\text{Number of data points}}\).
Since there are 7 data points, \(MAD=\frac{42}{7}=6\).

Answer:

• The mean hours students reported practicing their instruments is \(33\).
• The mean absolute deviation for the data is \(6\).