Core Measures and Formulas for Data Interpretation

The mean is the sum of all observations divided by their number. The median is the middle value of ordered data; with an even number of items it is the average of the two middle values. The mode is the value that occurs most often. Mean is affected by extreme values, median is not, and mode can be absent or have more than one value (bimodal).

\text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}}

\text{Weighted mean} = \frac{\sum w x}{\sum w}

For a frequency distribution, the mean is the sum of frequency times value, divided by total frequency. In a symmetric distribution, mean, median and mode are equal. For a moderately skewed distribution the empirical relation holds:

\text{Mode} = 3\\text{Median} - 2\\text{Mean}

If the distribution is positively skewed (long tail to the right), mean is greater than median, which is greater than mode. For negative skew, mode is greater than median, which is greater than mean.

For grouped data, the median class is the class where the cumulative frequency first reaches N / 2, and the modal class is the class with the highest frequency.

\text{Median} = L + \frac{N/2 - cf}{f} \times h

Here L is the lower limit of the median class, cf is the cumulative frequency before it, f is its frequency and h is its width.

Dispersion shows how spread out the data are. The range is the largest value minus the smallest. The quartile deviation is half the difference between the third and first quartiles. The mean deviation is the average of the absolute deviations from the mean or median. The variance is the average of squared deviations from the mean, and the standard deviation (SD) is its square root. The coefficient of variation (CV) is SD divided by mean, times 100, and is used to compare the consistency of two series: a smaller CV means more consistency.

A percentage expresses a number as a part of 100.

\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100

\text{Percentage change} = \frac{\text{New value} - \text{Old value}}{\text{Old value}} \times 100

The base of a percentage change is always the old (earlier) value. This is the most common source of wrong answers.

Percentage point versus percentage change. A rise in a rate from 20 per cent to 25 per cent is a rise of 5 percentage points (a simple difference) but a 25 per cent increase (5 / 20). When a question says "by how many percentage points", subtract. When it says "by what percentage", divide by the old value.

Successive changes. If a value changes by a per cent and then by b per cent, the net change is a + b + (a × b) / 100, with decreases entered as negatives. A rise of 20 per cent followed by a fall of 20 per cent gives 20 − 20 − 4 = −4 per cent, not zero.

Asymmetry of increase and decrease. If A is x per cent more than B, then B is x / (100 + x) × 100 per cent less than A. If A is 25 per cent more than B, then B is 20 per cent less than A.

A ratio a : b compares two quantities of the same unit and equals the fraction a / b. To divide a total in the ratio a : b, the first part is a / (a + b) of the total. A ratio has no units. When two ratios must be compared, convert them to decimals or percentages. A proportion states that two ratios are equal (a : b = c : d, so a × d = b × c).

The growth rate between two periods is the percentage change. The average annual growth over several years is not the simple average of the yearly rates unless a question explicitly asks for that. The compound annual growth rate (CAGR) gives the constant yearly rate that would take the starting value to the ending value:

\text{CAGR} = \left(\frac{\text{End value}}{\text{Start value}}\right)^{1/n} - 1

Here n is the number of years between the two values. The rule of 72 gives a quick estimate: the doubling time in years is about 72 divided by the yearly growth rate in per cent.

An index number measures change relative to a base period whose value is set to 100. If the index of a price is 130 in 2024 with 2020 as base, the price has risen 30 per cent since 2020. To compare two years that are both after the base, divide one index by the other, not subtract.

Example 1 (mean). Marks of five students are 12, 15, 18, 20 and 25. Mean = 90 / 5 = 18.

Example 2 (median, even count). Data: 4, 8, 10, 14. Median = (8 + 10) / 2 = 9.

Example 3 (mode from mean and median). Mean = 20 and median = 18. Mode = 3 × 18 − 2 × 20 = 14.

Example 4 (weighted mean). A student scores 80 in a paper of weight 2, 70 in a paper of weight 3 and 90 in a paper of weight 5. Weighted mean = (160 + 210 + 450) / 10 = 82.

Example 5 (combined mean). A class of 40 students has an average of 60 and another class of 60 students has an average of 70. Combined average = (40 × 60 + 60 × 70) / 100 = 66. Note that it is not the simple average of 60 and 70.

Example 6 (missing observation). The average of 5 numbers is 20. When a sixth number is added the average becomes 22. Sixth number = 6 × 22 − 5 × 20 = 32.

Example 7 (percentage). 45 of 180 students are girls. Percentage = 45 / 180 × 100 = 25 per cent.

Example 8 (percentage change). Enrolment rises from 400 to 500. Change = 100 / 400 × 100 = 25 per cent increase. If it falls from 500 to 400, the decrease is 100 / 500 × 100 = 20 per cent.

Example 9 (percentage points). Pass percentage rises from 60 to 66. The rise is 6 percentage points, and the percentage increase is 6 / 60 × 100 = 10 per cent.

Example 10 (successive change). A price is raised by 10 per cent and then by 20 per cent. Net rise = 10 + 20 + (10 × 20) / 100 = 32 per cent.

Example 11 (ratio split). A grant of ₹50,000 is divided between two departments in the ratio 3 : 2. First gets 3 / 5 × 50,000 = ₹30,000; second gets ₹20,000.

Example 12 (CAGR). A value grows from 100 to 144 in 2 years. CAGR = (1.44)^(1/2) − 1 = 1.2 − 1 = 20 per cent per year.

Example 13 (variance). Data: 2, 4, 6, 8. Mean = 5; squared deviations are 9, 1, 1, 9, which total 20; variance = 20 / 4 = 5 and SD ≈ 2.24.

Example 14 (compound growth). A population of 50,000 grows by 10 per cent per year for 2 years. Population = 50,000 × 1.1 × 1.1 = 60,500.