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Average and median

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Introduction to Average and Median

In quantitative aptitude, understanding how to summarize and interpret data is essential. Two fundamental measures used for this purpose are average and median. Both help us find a central or typical value in a set of numbers, but they do so in different ways.

Average, often called the arithmetic mean, is the sum of all values divided by the number of values. It gives a sense of the overall level of the data.

Median is the middle value when the data is arranged in order. It divides the data into two equal halves and is especially useful when the data contains extreme values or outliers.

For example, consider the monthly incomes (in INR) of five friends: 15,000; 18,000; 20,000; 22,000; and 1,00,000. The average income is pulled up by the one very high income, but the median gives a better sense of the typical income by focusing on the middle value.

In this chapter, we will explore how to calculate average and median, understand their differences, and learn when to use each measure effectively.

Average

The arithmetic mean or average of a set of numbers is found by adding all the numbers together and then dividing by how many numbers there are. This gives a single value representing the "central" or "typical" value of the data.

Formula:

Arithmetic Mean (Average)

\[\text{Average} = \frac{\sum_{i=1}^n x_i}{n}\]

Sum of all values divided by number of values

\(x_i\) = Each individual value
n = Total number of values

Let's see how this works with a simple example:

Student Marks (out of 100)
172
285
390
465
588
Total 400

To find the average marks:

  • Add all marks: 72 + 85 + 90 + 65 + 88 = 400
  • Divide by number of students: 400 / 5 = 80

So, the average marks scored by the students is 80.

Weighted Average

Sometimes, different values contribute unequally to the overall average. For example, if you buy 3 kg of apples at Rs.50 per kg and 2 kg of oranges at Rs.80 per kg, the average price per kg is not simply the average of 50 and 80. Instead, you use a weighted average, where each value is multiplied by its weight (quantity), then divided by the total weight.

Formula:

Weighted Average

\[\text{Weighted Average} = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}\]

Sum of products of values and weights divided by sum of weights

\(x_i\) = Values
\(w_i\) = Weights

This is very useful in real-life situations involving prices, quantities, or scores with different importance.

Median

The median is the middle value of a data set when the numbers are arranged in ascending order (from smallest to largest). It divides the data into two halves: half the numbers are below the median, and half are above.

Unlike average, the median is not affected much by extremely high or low values (called outliers). This makes it a better measure of central tendency when data is skewed.

Calculating Median

Before finding the median, always arrange the data in ascending order.

There are two cases:

  • Odd number of observations: Median is the middle value.
  • Even number of observations: Median is the average of the two middle values.
Median (Odd) Median (Even)

Formulas for Median:

Median for Odd Number of Observations

\[\text{Median} = x_{\left(\frac{n+1}{2}\right)}\]

Middle value in ordered data

n = Number of observations
x = Ordered data values

Median for Even Number of Observations

\[\text{Median} = \frac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2}\]

Average of two middle values in ordered data

n = Number of observations
x = Ordered data values

Worked Examples

Example 1: Calculating Average of Student Marks Easy
Five students scored the following marks in a test: 72, 85, 90, 65, and 88. Find their average marks.

Step 1: Add all the marks:

72 + 85 + 90 + 65 + 88 = 400

Step 2: Divide the sum by the number of students (5):

Average = 400 / 5 = 80

Answer: The average marks scored by the students is 80.

Example 2: Finding Median in an Odd Numbered Data Set Easy
The daily temperatures (in °C) recorded over 7 days are: 30, 25, 28, 32, 27, 29, 31. Find the median temperature.

Step 1: Arrange the temperatures in ascending order:

25, 27, 28, 29, 30, 31, 32

Step 2: Since there are 7 values (odd), median is the middle value at position \(\frac{7+1}{2} = 4\).

Step 3: The 4th value is 29.

Answer: The median temperature is 29°C.

Example 3: Finding Median in an Even Numbered Data Set Medium
A person's monthly expenses (in INR) for 6 months are: 12,000; 15,000; 14,000; 13,000; 16,000; 11,000. Find the median monthly expense.

Step 1: Arrange the expenses in ascending order:

11,000; 12,000; 13,000; 14,000; 15,000; 16,000

Step 2: Since there are 6 values (even), median is the average of the 3rd and 4th values:

Median = \(\frac{13,000 + 14,000}{2} = \frac{27,000}{2} = 13,500\)

Answer: The median monthly expense is Rs.13,500.

Example 4: Weighted Average Problem Medium
A shopkeeper sells 3 kg of rice at Rs.40 per kg and 5 kg of rice at Rs.50 per kg. Find the weighted average price per kg.

Step 1: Multiply each price by its weight (quantity):

3 x 40 = 120

5 x 50 = 250

Step 2: Add these products:

120 + 250 = 370

Step 3: Add the total weights:

3 + 5 = 8 kg

Step 4: Calculate weighted average price:

Weighted average = \(\frac{370}{8} = 46.25\)

Answer: The weighted average price per kg is Rs.46.25.

Example 5: Effect of Outliers on Average and Median Hard
The monthly salaries (in INR) of 7 employees in a company are: 20,000; 22,000; 21,000; 19,000; 23,000; 21,500; and 1,00,000. Find the average and median salary, and explain the effect of the outlier.

Step 1: Calculate the average:

Sum = 20,000 + 22,000 + 21,000 + 19,000 + 23,000 + 21,500 + 1,00,000 = 2,26,500

Average = \(\frac{2,26,500}{7} \approx 32,357\)

Step 2: Arrange salaries in ascending order:

19,000; 20,000; 21,000; 21,500; 22,000; 23,000; 1,00,000

Step 3: Since there are 7 values (odd), median is the 4th value:

Median = 21,500

Step 4: Interpretation:

The average salary is Rs.32,357, which is significantly higher due to the one very high salary of Rs.1,00,000 (an outlier). The median salary, Rs.21,500, better represents the typical salary of most employees as it is not affected by the outlier.

Answer: Average salary ≈ Rs.32,357; Median salary = Rs.21,500. Median is a better measure here due to the outlier.

Formula Bank

Arithmetic Mean (Average)
\[ \text{Average} = \frac{\sum_{i=1}^n x_i}{n} \]
where: \(x_i\) = individual values, \(n\) = number of values
Median for Odd Number of Observations
\[ \text{Median} = x_{\left(\frac{n+1}{2}\right)} \]
where: \(n\) = number of observations, \(x\) = ordered data values
Median for Even Number of Observations
\[ \text{Median} = \frac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2} \]
where: \(n\) = number of observations, \(x\) = ordered data values
Weighted Average
\[ \text{Weighted Average} = \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i} \]
where: \(x_i\) = values, \(w_i\) = weights

Tips & Tricks

Tip: Always arrange data in ascending order before calculating the median.

When to use: Before finding the median in any data set.

Tip: Use median instead of average when the data contains extreme values or outliers.

When to use: Data sets with very high or low values that can skew the average.

Tip: In weighted average problems, multiply each value by its weight before summing.

When to use: When different values contribute unequally to the overall average.

Tip: For large data sets, use the median position formulas to quickly find the median without counting manually.

When to use: When the number of observations is large.

Tip: Remember that average is sensitive to all values, while median depends only on the middle values.

When to use: To decide which measure better represents your data.

Common Mistakes to Avoid

❌ Not ordering data before finding the median.
✓ Always sort data in ascending order before calculating the median.
Why: Median depends on the position in ordered data; unordered data leads to incorrect results.
❌ Confusing median position formulas for odd and even number of values.
✓ Use the correct formula: middle value for odd, average of two middle values for even.
Why: Different formulas apply depending on whether the data count is odd or even.
❌ Ignoring weights in weighted average problems.
✓ Multiply each value by its weight and divide by total weight to get the correct weighted average.
Why: Weights affect the contribution of each value; ignoring them gives wrong results.
❌ Using average when data has extreme outliers.
✓ Use median to get a better measure of central tendency in such cases.
Why: Average is skewed by outliers, but median remains robust.
❌ Arithmetic errors when adding values for average calculation.
✓ Double-check addition and use a calculator if needed.
Why: Incorrect sums lead to wrong averages and wrong conclusions.
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