Mean Absolute Deviation worksheet

1. Mean absolute deviation around the median: 3.25

1. Sort the 8 values from smallest to largest, so the middle is easy to find

   $4\text{,}5\text{,}6\text{,}7\text{,}8\text{,}9\text{,}13\text{,}18$

2. With an even count, the median is halfway between the two middle values

   $\mathrm{median}$$=\frac{7+8}{2}$$=7.5$

3. Find how far each value is from the median, ignoring whether it is above or below, and add the distances

   $\text{|}4\text{−}7.5\text{|}+\text{|}7\text{−}7.5\text{|}+\text{…}+\text{|}18\text{−}7.5\text{|}$$=3.5+0.5+2.5+1.5+0.5+1.5+5.5+10.5$$=26$

4. Divide the total distance by the number of values to get the average distance

   $\mathrm{MAD}$$=\frac{26}{8}$$=3.25$

2. Median absolute deviation: 2.5

1. Sort the 8 values from smallest to largest, so the middle is easy to find

   $3\text{,}6\text{,}6\text{,}7\text{,}8\text{,}11\text{,}15\text{,}16$

2. With an even count, the median is halfway between the two middle values

   $\mathrm{median}$$=\frac{7+8}{2}$$=7.5$

3. Find how far each value is from the median, ignoring whether it is above or below, and add the distances

   $\text{|}3\text{−}7.5\text{|}+\text{|}6\text{−}7.5\text{|}+\text{…}+\text{|}16\text{−}7.5\text{|}$$=4.5+1.5+1.5+0.5+0.5+3.5+7.5+8.5$$=28$

4. Sort those distances so you can find their middle one

   $0.5\text{,}0.5\text{,}1.5\text{,}1.5\text{,}3.5\text{,}4.5\text{,}7.5\text{,}8.5$

5. The middle distance is the median absolute deviation

   $\mathrm{MAD}$$=\frac{1.5+3.5}{2}$$=2.5$

6. Optional: multiply by 1.4826 to estimate the standard deviation, if the data is roughly bell-shaped

   $\mathrm{\sigma ̂}$$=1.4826\times 2.5$$=3.706506$

3. Mean absolute deviation: 3.75

1. Find the mean (x̄): add the 8 values and divide by how many there are

   $\mathrm{x̄}$$=\frac{3+6+6+7+8+11+15+16}{8}$$=9$

2. Find how far each value is from the mean, ignoring whether it is above or below, and add the distances

   $\text{|}3\text{−}9\text{|}+\text{|}6\text{−}9\text{|}+\text{…}+\text{|}16\text{−}9\text{|}$$=6+3+3+2+1+2+6+7$$=30$

3. Divide the total distance by the number of values to get the average distance

   $\mathrm{MAD}$$=\frac{30}{8}$$=3.75$

4. Mean absolute deviation: 4.88

1. Find the mean (x̄): add the 8 values and divide by how many there are

   $\mathrm{x̄}$$=\frac{2+5+7+7+11+13+18+18}{8}$$=10.125$

2. Find how far each value is from the mean, ignoring whether it is above or below, and add the distances

   $\text{|}2\text{−}10.125\text{|}+\text{|}5\text{−}10.125\text{|}+\text{…}+\text{|}18\text{−}10.125\text{|}$$=8.125+5.125+3.125+3.125+0.875+2.875+7.875+7.875$$=39$

3. Divide the total distance by the number of values to get the average distance

   $\mathrm{MAD}$$=\frac{39}{8}$$=4.875$

5. Median absolute deviation: 1.5

1. Sort the 8 values from smallest to largest, so the middle is easy to find

   $3\text{,}5\text{,}6\text{,}6\text{,}7\text{,}8\text{,}15\text{,}16$

2. With an even count, the median is halfway between the two middle values

   $\mathrm{median}$$=\frac{6+7}{2}$$=6.5$

3. Find how far each value is from the median, ignoring whether it is above or below, and add the distances

   $\text{|}3\text{−}6.5\text{|}+\text{|}6\text{−}6.5\text{|}+\text{…}+\text{|}15\text{−}6.5\text{|}$$=3.5+0.5+0.5+1.5+0.5+1.5+9.5+8.5$$=26$

4. Sort those distances so you can find their middle one

   $0.5\text{,}0.5\text{,}0.5\text{,}1.5\text{,}1.5\text{,}3.5\text{,}8.5\text{,}9.5$

5. The middle distance is the median absolute deviation

   $\mathrm{MAD}$$=\frac{1.5+1.5}{2}$$=1.5$

6. Optional: multiply by 1.4826 to estimate the standard deviation, if the data is roughly bell-shaped

   $\mathrm{\sigma ̂}$$=1.4826\times 1.5$$=2.223903$

6. Mean absolute deviation: 3.38

1. Find the mean (x̄): add the 8 values and divide by how many there are

   $\mathrm{x̄}$$=\frac{3+5+7+6+10+11+11+16}{8}$$=8.625$

2. Find how far each value is from the mean, ignoring whether it is above or below, and add the distances

   $\text{|}3\text{−}8.625\text{|}+\text{|}5\text{−}8.625\text{|}+\text{…}+\text{|}16\text{−}8.625\text{|}$$=5.625+3.625+1.625+2.625+1.375+2.375+2.375+7.375$$=27$

3. Divide the total distance by the number of values to get the average distance

   $\mathrm{MAD}$$=\frac{27}{8}$$=3.375$

7. Mean absolute deviation around the median: 2.75

1. Sort the 8 values from smallest to largest, so the middle is easy to find

   $3\text{,}6\text{,}6\text{,}6\text{,}6\text{,}10\text{,}12\text{,}15$

2. With an even count, the median is halfway between the two middle values

   $\mathrm{median}$$=\frac{6+6}{2}$$=6$

3. Find how far each value is from the median, ignoring whether it is above or below, and add the distances

   $\text{|}3\text{−}6\text{|}+\text{|}6\text{−}6\text{|}+\text{…}+\text{|}12\text{−}6\text{|}$$=3+0+0+0+0+4+9+6$$=22$

4. Divide the total distance by the number of values to get the average distance

   $\mathrm{MAD}$$=\frac{22}{8}$$=2.75$

8. Mean absolute deviation: 3.66

1. Find the mean (x̄): add the 8 values and divide by how many there are

   $\mathrm{x̄}$$=\frac{3+5+6+9+8+12+18+12}{8}$$=9.125$

2. Find how far each value is from the mean, ignoring whether it is above or below, and add the distances

   $\text{|}3\text{−}9.125\text{|}+\text{|}5\text{−}9.125\text{|}+\text{…}+\text{|}12\text{−}9.125\text{|}$$=6.125+4.125+3.125+0.125+1.125+2.875+8.875+2.875$$=29.25$

3. Divide the total distance by the number of values to get the average distance

   $\mathrm{MAD}$$=\frac{29.25}{8}$$=3.65625$

9. Mean absolute deviation: 3.31

1. Find the mean (x̄): add the 8 values and divide by how many there are

   $\mathrm{x̄}$$=\frac{2+7+8+8+8+10+16+15}{8}$$=9.25$

2. Find how far each value is from the mean, ignoring whether it is above or below, and add the distances

   $\text{|}2\text{−}9.25\text{|}+\text{|}7\text{−}9.25\text{|}+\text{…}+\text{|}15\text{−}9.25\text{|}$$=7.25+2.25+1.25+1.25+1.25+0.75+6.75+5.75$$=26.5$

3. Divide the total distance by the number of values to get the average distance

   $\mathrm{MAD}$$=\frac{26.5}{8}$$=3.3125$

10. Median absolute deviation: 2

1. Sort the 8 values from smallest to largest, so the middle is easy to find

   $3\text{,}6\text{,}7\text{,}7\text{,}9\text{,}10\text{,}14\text{,}18$

2. With an even count, the median is halfway between the two middle values

   $\mathrm{median}$$=\frac{7+9}{2}$$=8$

3. Find how far each value is from the median, ignoring whether it is above or below, and add the distances

   $\text{|}3\text{−}8\text{|}+\text{|}6\text{−}8\text{|}+\text{…}+\text{|}18\text{−}8\text{|}$$=5+2+1+1+1+2+6+10$$=28$

4. Sort those distances so you can find their middle one

   $1\text{,}1\text{,}1\text{,}2\text{,}2\text{,}5\text{,}6\text{,}10$

5. The middle distance is the median absolute deviation

   $\mathrm{MAD}$$=\frac{2+2}{2}$$=2$

6. Optional: multiply by 1.4826 to estimate the standard deviation, if the data is roughly bell-shaped

   $\mathrm{\sigma ̂}$$=1.4826\times 2$$=2.965204$