Mean Absolute Deviation worksheet for calculating average deviation from the mean in data sets.

Let’s solve both problems step by step.

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Problem 1:

Data set:
72, 72, 74, 68, 99, 95, 44, 102, 73

Step 1: Find the mean (average) of the data.

Add all numbers:

72 + 72 = 144
144 + 74 = 218
218 + 68 = 286
286 + 99 = 385
385 + 95 = 480
480 + 44 = 524
524 + 102 = 626
626 + 73 = 699

There are 9 numbers.

Mean = 699 ÷ 9 = 77.666... → Let’s round to 77.7 for simplicity (but we’ll keep more decimals in calculation if needed).

Actually, let’s use exact value:
699 9 = 77.666... = 233/3 ≈ 77.6667

But since the problem says “round to nearest tenth”, we’ll use 77.7 as the mean for subtraction purposes? Wait — actually, better to use exact fraction or decimal during calculations and round only at the end.

Wait — let me recalculate sum carefully:

72
72 → 144
74 → 218
68 → 286
99 → 385
95 → 480
44 → 524
102 → 626
73 → 699 ✔

699 ÷ 9 = 77.666... → So mean = 77.7 when rounded to nearest tenth? But for accuracy in MAD, we should use the exact mean until final step.

Actually, standard practice is to use the exact mean (even if repeating) for subtracting from each data point, then round the final MAD.

So let’s use mean = 699/9 = 77.666... = 77.6667 (we’ll use 77.6667 for calculations)

Step 2: Subtract mean from each data point and take absolute value.

List of data points:
72, 72, 74, 68, 99, 95, 44, 102, 73

Compute |data - mean|:

1. |72 - 77.6667| = |-5.6667| = 5.6667
2. |72 - 77.6667| = 5.6667
3. |74 - 77.6667| = |-3.6667| = 3.6667
4. |68 - 77.6667| = |-9.6667| = 9.6667
5. |99 - 77.6667| = 21.3333
6. |95 - 77.6667| = 17.3333
7. |44 - 77.6667| = |-33.6667| = 33.6667
8. |102 - 77.6667| = 24.3333
9. |73 - 77.6667| = |-4.6667| = 4.6667

Now add these absolute deviations:

5.6667
+5.6667 = 11.3334
+3.6667 = 15.0001
+9.6667 = 24.6668
+21.3333 = 46.0001
+17.3333 = 63.3334
+33.6667 = 97.0001
+24.3333 = 121.3334
+4.6667 = 126.0001

Sum of absolute deviations ≈ 126.0

Step 3: Divide by number of data points (9):

MAD = 126.0 ÷ 9 = 14.0

✔ That’s clean! Probably because the total was exactly 126.

Let me verify with fractions to be sure.

Mean = 699/9 = 233/3

Each deviation:

72 - 233/3 = (216 - 233)/3 = -17/3 → abs = 17/3
Same for second 72 → 17/3
74 - 233/3 = (222 - 233)/3 = -11/3 → abs = 11/3
68 - 233/3 = (204 - 233)/3 = -29/3 → abs = 29/3
99 - 233/3 = (297 - 233)/3 = 64/3 → abs = 64/3
95 - 233/3 = (285 - 233)/3 = 52/3 → abs = 52/3
44 - 233/3 = (132 - 233)/3 = -101/3 → abs = 101/3
102 - 233/3 = (306 - 233)/3 = 73/3 → abs = 73/3
73 - 233/3 = (219 - 233)/3 = -14/3 → abs = 14/3

Now sum numerators:

17 + 17 = 34
+11 = 45
+29 = 74
+64 = 138
+52 = 190
+101 = 291
+73 = 364
+14 = 378

Total sum of absolute deviations = 378 / 3 = 126

Then MAD = 126 ÷ 9 = 14.0

Perfect.

So Problem 1 answer: 14.0

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Problem 2:

Test scores:
65%, 70%, 85%, 89%, 90%
55%, 100%, 100%

That’s 8 data points.

Step 1: Find the mean.

Add them:

65 + 70 = 135
135 + 85 = 220
220 + 89 = 309
309 + 90 = 399
399 + 55 = 454
454 + 100 = 554
554 + 100 = 654

Number of values = 8

Mean = 654 ÷ 8 = 81.75

Step 2: Find absolute deviation from mean for each.

|65 - 81.75| = 16.75
|70 - 81.75| = 11.75
|85 - 81.75| = 3.25
|89 - 81.75| = 7.25
|90 - 81.75| = 8.25
|55 - 81.75| = 26.75
|100 - 81.75| = 18.25
|100 - 81.75| = 18.25

Now sum these:

16.75
+11.75 = 28.50
+3.25 = 31.75
+7.25 = 39.00
+8.25 = 47.25
+26.75 = 74.00
+18.25 = 92.25
+18.25 = 110.50

Sum of absolute deviations = 110.5

Step 3: Divide by 8

MAD = 110.5 ÷ 8 = 13.8125

Round to nearest tenth → 13.8

Check:

110.5 ÷ 8:

8 × 13 = 104
110.5 - 104 = 6.5
6.5 ÷ 8 = 0.8125 → so 13.8125 → rounds to 13.8

✔ Correct.

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Final Answer:
Problem 1: 14.0
Problem 2: 13.8