Geometric and Harmonic Mean worksheet

1. Average change per period (geometric): 6.57%

1. Turn each percent change into a growth factor, the number you multiply by for that change

   $\text{14\% → 1.14, -6\% → 0.94, 18\% → 1.18, 2\% → 1.02}$

2. Multiply the 4 factors together, since they work by multiplying

   $1.14\times 0.94\times 1.18\times 1.02$$=1.28977776$

3. Take the 4th root of the product: the one number that, used 4 times, gives the same product

   $\mathrm{GM}$$={1.28977776}^{\frac{1}{4}}$$=1.0656846763$

4. Subtract 1 and turn it into a percent. That is the steady change per period (the compound rate)

   $\left(1.0656846763-1\right)\times 100$$=\mathrm{6.568468\%}$

5. Check: repeating that rate every period gives the same total change

   $\text{total change =}\left({1.0656846763}^4-1\right)\times 100$$=\mathrm{28.977776\%}$

2. Geometric mean: 6

1. Multiply the 3 values together, since they work by multiplying

   $3\times 12\times 6$$=216$

2. Take the cube root of the product: the one number that, used 3 times, gives the same product

   $\mathrm{GM}$$={216}^{\frac{1}{3}}$$=6$

3. Harmonic mean: add up 1 divided by each value, then divide the count by that total

   $\mathrm{HM}$$=\frac{3}{\frac{1}{3}+\frac{1}{12}+\frac{1}{6}}$$=5.1428571429$

4. For comparison, the ordinary (arithmetic) mean: the sum divided by the count

   $\mathrm{AM}$$=\frac{3+12+6}{3}$$=7$

5. For positive values, the ordinary mean is never below the geometric mean, and the harmonic mean is never above it

   $\text{7 ≥ 6 ≥ 5.142857}$

6. So the geometric mean, which sits between the other two, is

   $\mathrm{GM}$$=6$

3. Geometric mean: 5.83

1. Multiply the 3 values together, since they work by multiplying

   $3\times 11\times 6$$=198$

2. Take the cube root of the product: the one number that, used 3 times, gives the same product

   $\mathrm{GM}$$={198}^{\frac{1}{3}}$$=5.8284766833$

3. Harmonic mean: add up 1 divided by each value, then divide the count by that total

   $\mathrm{HM}$$=\frac{3}{\frac{1}{3}+\frac{1}{11}+\frac{1}{6}}$$=5.0769230769$

4. For comparison, the ordinary (arithmetic) mean: the sum divided by the count

   $\mathrm{AM}$$=\frac{3+11+6}{3}$$=6.6666666667$

5. For positive values, the ordinary mean is never below the geometric mean, and the harmonic mean is never above it

   $\text{6.666667 ≥ 5.828477 ≥ 5.076923}$

6. So the geometric mean, which sits between the other two, is

   $\mathrm{GM}$$=5.8284766833$

4. Average change per period (geometric): 6.35%

1. Turn each percent change into a growth factor, the number you multiply by for that change

   $\text{10\% → 1.1, -5\% → 0.95, 20\% → 1.2, 2\% → 1.02}$

2. Multiply the 4 factors together, since they work by multiplying

   $1.1\times 0.95\times 1.2\times 1.02$$=1.27908$

3. Take the 4th root of the product: the one number that, used 4 times, gives the same product

   $\mathrm{GM}$$={1.27908}^{\frac{1}{4}}$$=1.0634680016$

4. Subtract 1 and turn it into a percent. That is the steady change per period (the compound rate)

   $\left(1.0634680016-1\right)\times 100$$=\mathrm{6.3468\%}$

5. Check: repeating that rate every period gives the same total change

   $\text{total change =}\left({1.0634680016}^4-1\right)\times 100$$=\mathrm{27.908\%}$

5. Geometric mean: 5.19

1. Multiply the 3 values together, since they work by multiplying

   $2\times 10\times 7$$=140$

2. Take the cube root of the product: the one number that, used 3 times, gives the same product

   $\mathrm{GM}$$={140}^{\frac{1}{3}}$$=5.1924941019$

3. Harmonic mean: add up 1 divided by each value, then divide the count by that total

   $\mathrm{HM}$$=\frac{3}{\frac{1}{2}+\frac{1}{10}+\frac{1}{7}}$$=4.0384615385$

4. For comparison, the ordinary (arithmetic) mean: the sum divided by the count

   $\mathrm{AM}$$=\frac{2+10+7}{3}$$=6.3333333333$

5. For positive values, the ordinary mean is never below the geometric mean, and the harmonic mean is never above it

   $\text{6.333333 ≥ 5.192494 ≥ 4.038462}$

6. So the geometric mean, which sits between the other two, is

   $\mathrm{GM}$$=5.1924941019$

6. Average change per period (geometric): 5.62%

1. Turn each percent change into a growth factor, the number you multiply by for that change

   $\text{12\% → 1.12, -7\% → 0.93, 16\% → 1.16, 3\% → 1.03}$

2. Multiply the 4 factors together, since they work by multiplying

   $1.12\times 0.93\times 1.16\times 1.03$$=1.24450368$

3. Take the 4th root of the product: the one number that, used 4 times, gives the same product

   $\mathrm{GM}$$={1.24450368}^{\frac{1}{4}}$$=1.0562070118$

4. Subtract 1 and turn it into a percent. That is the steady change per period (the compound rate)

   $\left(1.0562070118-1\right)\times 100$$=\mathrm{5.620701\%}$

5. Check: repeating that rate every period gives the same total change

   $\text{total change =}\left({1.0562070118}^4-1\right)\times 100$$=\mathrm{24.450368\%}$

7. Geometric mean: 6.21

1. Multiply the 3 values together, since they work by multiplying

   $4\times 10\times 6$$=240$

2. Take the cube root of the product: the one number that, used 3 times, gives the same product

   $\mathrm{GM}$$={240}^{\frac{1}{3}}$$=6.2144650119$

3. Harmonic mean: add up 1 divided by each value, then divide the count by that total

   $\mathrm{HM}$$=\frac{3}{\frac{1}{4}+\frac{1}{10}+\frac{1}{6}}$$=5.8064516129$

4. For comparison, the ordinary (arithmetic) mean: the sum divided by the count

   $\mathrm{AM}$$=\frac{4+10+6}{3}$$=6.6666666667$

5. For positive values, the ordinary mean is never below the geometric mean, and the harmonic mean is never above it

   $\text{6.666667 ≥ 6.214465 ≥ 5.806452}$

6. So the geometric mean, which sits between the other two, is

   $\mathrm{GM}$$=6.2144650119$

8. Geometric mean: 5.24

1. Multiply the 3 values together, since they work by multiplying

   $3\times 12\times 4$$=144$

2. Take the cube root of the product: the one number that, used 3 times, gives the same product

   $\mathrm{GM}$$={144}^{\frac{1}{3}}$$=5.2414827884$

3. Harmonic mean: add up 1 divided by each value, then divide the count by that total

   $\mathrm{HM}$$=\frac{3}{\frac{1}{3}+\frac{1}{12}+\frac{1}{4}}$$=4.5$

4. For comparison, the ordinary (arithmetic) mean: the sum divided by the count

   $\mathrm{AM}$$=\frac{3+12+4}{3}$$=6.3333333333$

5. For positive values, the ordinary mean is never below the geometric mean, and the harmonic mean is never above it

   $\text{6.333333 ≥ 5.241483 ≥ 4.5}$

6. So the geometric mean, which sits between the other two, is

   $\mathrm{GM}$$=5.2414827884$

9. Average change per period (geometric): 4.03%

1. Turn each percent change into a growth factor, the number you multiply by for that change

   $\text{9\% → 1.09, -6\% → 0.94, 11\% → 1.11, 3\% → 1.03}$

2. Multiply the 4 factors together, since they work by multiplying

   $1.09\times 0.94\times 1.11\times 1.03$$=1.17142518$

3. Take the 4th root of the product: the one number that, used 4 times, gives the same product

   $\mathrm{GM}$$={1.17142518}^{\frac{1}{4}}$$=1.0403480052$

4. Subtract 1 and turn it into a percent. That is the steady change per period (the compound rate)

   $\left(1.0403480052-1\right)\times 100$$=\mathrm{4.034801\%}$

5. Check: repeating that rate every period gives the same total change

   $\text{total change =}\left({1.0403480052}^4-1\right)\times 100$$=\mathrm{17.142518\%}$

10. Geometric mean: 6.46

1. Multiply the 3 values together, since they work by multiplying

   $3\times 15\times 6$$=270$

2. Take the cube root of the product: the one number that, used 3 times, gives the same product

   $\mathrm{GM}$$={270}^{\frac{1}{3}}$$=6.4633040701$

3. Harmonic mean: add up 1 divided by each value, then divide the count by that total

   $\mathrm{HM}$$=\frac{3}{\frac{1}{3}+\frac{1}{15}+\frac{1}{6}}$$=5.2941176471$

4. For comparison, the ordinary (arithmetic) mean: the sum divided by the count

   $\mathrm{AM}$$=\frac{3+15+6}{3}$$=8$

5. For positive values, the ordinary mean is never below the geometric mean, and the harmonic mean is never above it

   $\text{8 ≥ 6.463304 ≥ 5.294118}$

6. So the geometric mean, which sits between the other two, is

   $\mathrm{GM}$$=6.4633040701$