Harmonic Mean as Central Tendency

In this class, We discuss Harmonic Mean as Central Tendency.

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The reader should have prior knowledge of geometric mean. Click Here.

The reader will understand harmonic mean and the situations where harmonic mean helps.

Harmonic Mean:

The harmonic mean is calculated by dividing the number of values in the data series by the sum of reciprocals of each data value.

Given n data values.

x1, x2, x3, . . ., xn.

Harmonic Mean = n/((1/x1) +(1/x2) +. .+(1/xn)).

The harmonic mean is useful for data that involves rates—I.e. Price per earning and miles per hour.

Example:

We sell milk at 8, 10, 12, and 15 rupees per litre.

Find the average price of rupees per litre.

HM = n/((1/x1) +(1/x2) +. .+(1/xn)).

HM = 4/((1/4) +(1/8) +(1/12)+(1/15)).

HM = 10.667

Example 2: Weighted Harmonic Mean

WHM = Σwi/(Σ(wi/xi))

The below example will help the reader understand the weighted harmonic mean.

Example:

The index of a stock of company A and company B.

Company A reports a marketing capital of one billion and earnings of 20 million.

Company B reports a marketing capital of 20 billion and earnings of 5 billion.

The investment contains 40% of company A and 60% of company B.

Find the average?

Investment percentages are weights in our example.

price/earnings = 1billion / 20 million.

convert the million to billion.

P/E = 1billion/ 20 million = 50

P/E = 20 billion/5 billion = 4

X1 = 50 and X2 = 4

W1 = 0.4 and W2 = 0.6

Harmonic mean = (0.4 + 0.6)/((0.4/50) + (0.6/4))

HM = 6.33

If we use weighted arithmetic mean, we get a value not suitable.

Weighted Arithmetic Mean = 0.4 * 50 + 0.6 * 4 = 22.4