What Each Average Means
In statistics, the word average does not always mean only one thing. Different datasets need different kinds of averages depending on whether we want a simple center, a middle position, a most common value, or a rate-based summary.
1. Arithmetic Mean
This is the usual average: add all values and divide by how many values there are.
It uses every observation, so it gives a balanced overall center.
It works best when the data is fairly regular and extreme values are not dominating the set.
Application: Average exam score, average rainfall, average monthly sales.
2. Median
Median is the middle value after arranging the data from smallest to largest.
It is useful when the dataset has outliers because it is not pulled strongly by one very high or very low value.
Application: Income, house prices, or waiting times where a few extreme values may distort the mean.
3. Mode
Mode is the value that appears most often.
It is the best average when the most common category or repeated observation matters more than a numerical balance point.
Application: Most common shoe size sold, most chosen transport option, or most frequent defect type.
4. Weighted Mean
Weighted mean is used when some values deserve more importance than others.
Each value is multiplied by its weight, then we divide by the total weight.
It is more realistic than the simple mean when contributions are unequal.
Application: Grade point calculation, index numbers, and average price when quantities purchased are different.
5. Geometric Mean
Geometric mean is useful for values that grow or shrink by percentages or ratios.
Instead of adding values, it combines multiplicative change.
It gives a steadier picture of long-term growth than the arithmetic mean.
Application: Investment returns, population growth, inflation trends, and business growth rates.
6. Harmonic Mean
Harmonic mean is appropriate when averaging rates, especially when the same distance, work, or quantity is involved.
It gives more weight to smaller values, which is important in speed and rate problems.
Application: Average speed, average cost per unit across equal quantities, or average productivity rates.
Worked Examples
1. Arithmetic Mean
Marks = 60, 70, 80, 90
Mean = (60 + 70 + 80 + 90) / 4 = 300 / 4 = 75
2. Median
Incomes = 20, 22, 24, 26, 120
Ordered data already shown.
Middle value = 24
Here the very high income 120 changes the mean a lot, but the median stays at the central position.
3. Mode
Shoe sizes sold = 6, 7, 7, 7, 8, 9
Most frequent value = 7
So size 7 is the mode and the most commonly demanded size.
4. Weighted Mean
A student scores 80 in a test worth 40% and 90 in a test worth 60%.
Weighted mean = (80 x 40 + 90 x 60) / 100
= (3200 + 5400) / 100 = 86
Another weighted mean example
A shop buys 2 kg rice at Rs. 40 per kg and 3 kg rice at Rs. 50 per kg.
Weighted mean price = (2 x 40 + 3 x 50) / (2 + 3)
= (80 + 150) / 5 = Rs. 46 per kg
Here quantity acts as the weight, so the bigger purchase influences the average more.
One more weighted mean example
A final grade is based on Assignment 20%, Midterm 30%, Final Exam 50%.
Scores are 70, 80, 90.
Weighted mean = (70 x 20 + 80 x 30 + 90 x 50) / 100
= (1400 + 2400 + 4500) / 100 = 83
This shows why a simple average would be wrong when parts do not carry equal importance.
5. Geometric Mean
An investment grows by factors 1.10 and 1.20 in two years.
Geometric mean growth factor = sqrt(1.10 x 1.20)
= sqrt(1.32) = about 1.149
So the average growth rate is about 14.9% per year.
Another geometric mean example
A town's population changes by 5% in one year and 15% in the next.
Growth factors are 1.05 and 1.15.
Geometric mean = sqrt(1.05 x 1.15) = sqrt(1.2075) = about 1.099
So the average annual growth rate is about 9.9%, not simply 10% by rough addition.
Why not arithmetic mean here?
If a value rises 50% and then falls 50%, arithmetic averaging suggests 0% average change,
but the original value does not come back. Geometric mean captures such multiplicative change more correctly.
6. Harmonic Mean
A car travels one half of a trip at 60 km/h and the other half at 40 km/h.
Harmonic mean = 2ab / (a + b)
= 2 x 60 x 40 / (60 + 40)
= 4800 / 100 = 48 km/h
Another harmonic mean example
A worker completes the same job once at 6 hours and another time at 3 hours.
The work rates are 1/6 and 1/3 job per hour.
Harmonic mean of the times = 2 x 6 x 3 / (6 + 3) = 36 / 9 = 4 hours
Harmonic mean is used because the same amount of work is being compared at different rates.
One more harmonic mean example
If equal money is invested in two funds with P/E ratios 10 and 20, the average P/E ratio is not 15.
Harmonic mean = 2 x 10 x 20 / (10 + 20) = 400 / 30 = about 13.33
This is another case where averaging rates or ratios needs harmonic mean, not arithmetic mean.
Notice that each example answers a different practical question. That is why choosing the correct average matters more than using the most familiar one.