### What is harmonic mean in statistics?

The harmonic mean is one of the several types of “mean”. It is the reciprocal of the arithmetic mean. To calculate it we usually divide the sum of the reciprocals of all values by the total number of terms.

### Formula of Harmonic mean:

Generally, for the ungroup data we use the following formula:

In the above formula:

- “Σ (1 / X
_{i})” is the sum of reciprocals of all values - “n” is the total number of terms

### Advantages and disadvantages of harmonic mean:

In this section, we’ll discuss some basic pros and cons of harmonic mean.

**Advantages: **

- The mathematical formula thoroughly defines it
- It is based on all the observations
- It is not much affected by the sampling stability

**Disadvantages: **

- It is not easily understandable
- It cannot calculate if any observation is zero
- It gives high weightage to small values.

### Examples:

In this section, we’ll cover the basic method to calculate the harmonic mean of any ungroup data.

**Example 1: **

Evaluate the harmonic mean of the following data:

1, 5, 7, 3, 9, 12, 13

**Solution: **

**Step 1:** Extract the data

Let X = 1, 5, 7, 3, 9, 12, 13

Total number of terms = n = 7

**Step 2: **Put the values in the formula.

Harmonic mean = n / Σ (1 / X_{i})

Harmonic mean = 7 / (1 / 1 + 1 / 5 + 1 / 7 + 1 / 3 + 1 / 9 + 1 / 12 + 1 / 13)

Harmonic mean = 7 / (1 + 0.2 + 0.1429 + 0.3333 + 0.1111 + 0.0833 + 0.0769)

Harmonic mean = 7 / 1.9475

**Harmonic mean = 3.59424**

**Example 2: **

If a person is driving a car at the rate of 50 km/h during the first 30 kilometers, at 60 km/h during the 2^{nd} 30 kilometers, 90km/h during the 3^{rd} 30 kilometers, and 100km/h during the 4^{th} 30 kilometers calculate the average speed of the car.

**Solution: **

**Step 1:** Extract the data

Let,

Speed of car = X = 50, 60, 90, 100

Total number of terms = n = 4

**Step 2: **Put the values in the formula.

Harmonic mean = n / Σ (1 / X_{i})

Harmonic mean = 4 / (1 / 50 + 1 / 60 + 1 / 90 + 1 / 100)

Harmonic mean = 4 / (0.02 + 0.0167 + 0.0111 + 0.01)

Harmonic mean = 4 / 0.0578

**Harmonic mean = 69.23077**