4.3 Elementary Statistics & Data Interpretation

Measures of Central Tendency

Mean (average). For ungrouped data, mean = (sum of observations) / (number of observations). For grouped data, multiply each class midpoint x_i by its frequency f_i, sum, and divide by total frequency N: mean = Σ(f_i × x_i) / N.

Median. Sort the data; the median is the middle value for an odd count and the average of the two middle values for an even count. For grouped data, use the formula median = L + ((N/2 - CF)/f) × h, where L is the lower limit of the median class, CF is the cumulative frequency before it, f is its frequency, and h is the class width.

Mode. The most frequently occurring value. For grouped data, mode = L + ((f1 - f0)/(2f1 - f0 - f2)) × h, where L is the lower limit of the modal class, f1 is its frequency, f0 the previous class frequency, f2 the next class frequency, and h the class width.

Empirical relation for moderately skewed distributions: 3 × median = 2 × mean + mode. Use it as a quick check or to estimate one measure from the other two.

Range = maximum − minimum. The simplest measure of dispersion.

Worked Examples

Worked example 1 (ungrouped). Find the mean, median, and mode of: 12, 15, 18, 15, 22, 15, 9.

• Sum = 12 + 15 + 18 + 15 + 22 + 15 + 9 = 106. Count = 7. Mean = 106/7 ≈ 15.14.
• Sorted: 9, 12, 15, 15, 15, 18, 22. Median (4th value) = 15.
• Mode = 15 (appears three times).
• Range = 22 − 9 = 13.

Worked example 2 (grouped mean). A class has 20 students with marks grouped as below. Estimate the mean.

Mean = 780 / 20 = 39.

Worked example 3 (median, even count). Find the median of 8, 3, 5, 11, 7, 4.

Sort: 3, 4, 5, 7, 8, 11. Even count (6), so take the average of the 3rd and 4th values: (5 + 7)/2 = 6.

Reading Bar Graphs and Histograms

A bar graph compares discrete categories; bars have gaps. A histogram displays continuous class intervals; bars touch. In a histogram, frequency may be shown directly or as frequency density (frequency / class width) when class widths differ.

To answer an RRB bar-graph question:

1. Read the title and units first.
2. Note the scale on each axis.
3. Compute totals, differences, or ratios from the bars — do not estimate visually.
4. Watch for double-bar (grouped) charts comparing two years or categories.

Reading Pie Charts

A pie chart represents parts of a whole as sectors of a circle. The whole is 360° and the whole quantity. Each slice's angle = (slice value / total) × 360°. Conversely, slice value = (slice angle / 360°) × total.

Worked example 4 (pie chart). A pie chart shows a railway budget of ₹72,000 crore split into four sectors. If "Infrastructure" occupies 150° of the pie, what is its allocation?

Infrastructure value = (150 / 360) × 72,000 = (5/12) × 72,000 = ₹30,000 crore.

Data Interpretation from Tables

RRB often presents a table of values across rows (years, regions, categories) and columns (years, products) and asks for totals, percentages, ratios, or growth. The efficient approach:

• Identify exactly which cells the question needs.
• Compute totals with column/row sums; do not add in your head if more than four numbers are involved.
• For percentage change: ((new - old) / old) × 100. Memorise common fractions: 1/2 = 50%, 1/3 ≈ 33.3%, 1/4 = 25%, 1/5 = 20%, 1/6 ≈ 16.7%.
• For ratios, simplify by dividing both sides by their greatest common divisor; for 24:36, divide by 12 to get 2:3.

Common Traps in Data Interpretation

• Wrong total: pie chart values must sum to the total given; check before you compute a percentage.
• Mixed units: a table in lakhs of rupees with a question in crores — convert (1 crore = 100 lakh) before computing.
• Histogram with unequal class widths: the taller bar may not mean the larger frequency; use frequency density.
• Approximation errors: when the answer choices are close, keep one extra decimal through intermediate steps.
• "Approximate" vs "exact" wording: the question will say which it wants; do not round unless instructed.

Speed Tips

• Memorise 1/7 ≈ 0.143, 1/8 = 0.125, 1/9 ≈ 0.111, 1/11 ≈ 0.091 — they recur in DI percentages.
• For an average of evenly spaced numbers (e.g., 10, 20, 30, ..., 100), the mean equals the median equals (first + last)/2.
• To compute a percentage of a round number quickly: X% of Y = (X × Y)/100; mentally shift the decimal — 18% of 250 = 18 × 2.5 = 45.
• For growth over two years at r% and s%, the combined rate ≈ r + s + (r × s)/100 when both rates are small.