Statistics for NDA

Statistics is the friendliest chapter in the NDA maths paper. The arithmetic is light, the formulas are few, and many questions test one idea: what happens to the mean or the standard deviation when every value is changed. The syllabus covers classification of data, frequency and cumulative frequency distributions, histograms, pie charts and frequency polygons, mean, median and mode, variance and standard deviation, and correlation and regression.

That last item surprises people who studied statistics only in Class 10. Regression questions are short and formula-driven, so learn the four facts below and they become easy marks.

Typical question types are:

• the mean, median or mode of a small data set, or the effect of changing one value;
• the median or mode of grouped data using the class-interval formulas;
• variance and SD of a set, often a set of consecutive numbers;
• what happens to the mean, median, SD or variance when every value is shifted or scaled;
• combined mean of two groups;
• pie-chart angles and reading a histogram or ogive;
• regression coefficients, the correlation coefficient and the point where regression lines meet.

Measures of central tendency

• Mean = sum of values ÷ number of values. For a frequency table, mean = Σfx/Σf.
• Median = the middle value of the ordered data. For n values, it is the ((n + 1)/2)th value when n is odd, and the average of the (n/2)th and (n/2 + 1)th values when n is even.
• Mode = the most frequent value. A set can have more than one mode.
• Empirical relation for moderately skewed data: mode ≈ 3 median − 2 mean.

Handy results: the mean of the first n natural numbers is (n + 1)/2. The combined mean of two groups with sizes n₁ and n₂ and means x̄₁ and x̄₂ is (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂).

Grouped data

For a continuous frequency distribution with class width h:

• Median = l + [(N/2 − cf)/f] × h, where l is the lower limit of the median class, N the total frequency, cf the cumulative frequency before the median class, and f the median class's frequency.
• Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where f₁ is the modal class frequency and f₀, f₂ are the frequencies of the classes just before and after it.

The median class is the first class whose cumulative frequency reaches N/2. The modal class is the one with the highest frequency.

Graphs

• A histogram shows a continuous distribution as touching bars; a frequency polygon joins the midpoints of their tops.
• An ogive is the cumulative frequency curve. The less-than and more-than ogives meet at the median.
• In a pie chart, the angle of a sector = (value ÷ total) × 360°.

Measures of spread

• Range = largest − smallest value.
• Variance σ² = mean of the squared deviations from the mean = Σ(x − x̄)²/n. An equivalent shortcut is σ² = (Σx²/n) − x̄².
• Standard deviation σ = √variance.
• Coefficient of variation (CV) = (σ/mean) × 100. The data set with the lower CV is the more consistent one.
• The variance of the first n natural numbers is (n² − 1)/12.

The SD is zero only when every value is the same.

Shifting and scaling

This table answers a large share of NDA statistics questions.

So multiplying by −2 doubles the SD (not −2 times it) and makes the variance four times as large.

Correlation and regression

The correlation coefficient r measures how closely two variables follow a straight line.

• −1 ≤ r ≤ 1. r = 1 is perfect positive correlation, r = −1 perfect negative, and r = 0 means no linear correlation.
• r = cov(x, y)/(σxσy).
• r does not change when you shift the origin or change the scale of either variable, as long as the scale factors are positive.

There are two regression lines: y on x (used to estimate y) and x on y (used to estimate x).

• The regression coefficient of y on x is byx = r(σy/σx), and of x on y is bxy = r(σx/σy).
• r² = byx × bxy, and r has the same sign as the two coefficients (which always share a sign).
• Both lines pass through the point of means (x̄, ȳ). Solving the two line equations together gives the means.
• If r = ±1, the two lines coincide.

Worked NDA-style MCQs

Q1. The mean of 5 observations is 12. One observation, 20, is replaced by 10. The new mean is:
(a) 11 (b) 10 (c) 12 (d) 9

The total was 5 × 12 = 60. The new total is 60 − 20 + 10 = 50, so the mean is 10. Answer: (b).

Q2. For the distribution below, the median class and the median are:

(a) 20–30; 25.8 (approx.) (b) 20–30; 25 (c) 30–40; 32.5 (d) 10–20; 18.3

N = 40 and N/2 = 20. The cumulative frequencies are 5, 13, 25, …, so the median class is 20–30, with l = 20, cf = 13, f = 12 and h = 10. Median = 20 + (7/12) × 10 = 20 + 5.83 = 25.83. Answer: (a).

Q3. The variance of the first 10 natural numbers is:
(a) 9.25 (b) 5.5 (c) 8.25 (d) 2.87

(n² − 1)/12 = 99/12 = 8.25. Option (d) is the SD, √8.25 ≈ 2.87. Answer: (c).

Q4. If byx = 0.8 and bxy = 0.2, the correlation coefficient is:
(a) 0.16 (b) 0.64 (c) 1.0 (d) 0.4

r² = 0.8 × 0.2 = 0.16. Both coefficients are positive, so r = +0.4. Answer: (d).

Q5. The two regression lines are x + 2y = 5 and 2x + 3y = 8. The means x̄ and ȳ are:
(a) 1, 2 (b) 2, 1 (c) 3, 1 (d) 1, 3

Both lines pass through (x̄, ȳ). From the first, x = 5 − 2y; substituting, 10 − 4y + 3y = 8, so y = 2 and x = 1. Answer: (a).

Q6. In a class, 20 students have a mean mark of 60 and 30 students have a mean mark of 70. The mean of all 50 is:
(a) 64 (b) 65 (c) 66 (d) 67

(20 × 60 + 30 × 70)/50 = (1,200 + 2,100)/50 = 66. The answer is pulled towards the larger group. Answer: (c).

Common mistakes

• Finding the median without sorting the data first.
• Scaling the variance by k instead of k². The SD scales by the size of k; the variance by k².
• Averaging two means directly when the groups have different sizes.
• Taking r = √(byx × bxy) as always positive. r takes the sign of the regression coefficients.
• Mixing up the two regression lines. To estimate y from x, use the line of y on x.

Practice set

1. The mean of x, x + 2, x + 4, x + 6 and x + 8 is: (a) x + 2 (b) x + 4 (c) x + 5 (d) 5x
2. If the mean is 30 and the median 28, the mode by the empirical relation is: (a) 24 (b) 26 (c) 32 (d) 29
3. The variance of 2, 4, 6, 8, 10 is: (a) 4 (b) 6 (c) 8 (d) 10
4. The variance of a set is 5. Each value is multiplied by −2. The new variance is: (a) 20 (b) 10 (c) −10 (d) −20
5. If r = 0.6, σx = 2 and σy = 3, then byx is: (a) 0.4 (b) 0.6 (c) 1.2 (d) 0.9
6. A data set has mean 40 and SD 8. Its coefficient of variation is: (a) 5% (b) 20% (c) 32% (d) 48%
7. In a pie chart, an item that is one-sixth of the total has a sector angle of: (a) 30° (b) 45° (c) 60° (d) 72°
8. The median of 10, 3, 7, 15, 12, 8 is: (a) 8 (b) 9.5 (c) 10 (d) 9

Answers:

1. (b). The sum is 5x + 20, and dividing by 5 gives x + 4.
2. (a). 3 × 28 − 2 × 30 = 84 − 60 = 24.
3. (c). The mean is 6; squared deviations 16, 4, 0, 4, 16 sum to 40; 40/5 = 8.
4. (a). Variance scales by (−2)² = 4.
5. (d). 0.6 × 3/2 = 0.9.
6. (b). 8/40 × 100 = 20%.
7. (c). 360°/6 = 60°.
8. (d). Sorted: 3, 7, 8, 10, 12, 15; the middle two average to 9.

What to do next

• Rebuild the shifting-and-scaling table from memory until it comes out right without a pause.
• Work one grouped-data question for the median and one for the mode each day for a week.
• Learn the four regression facts (r² = byx × bxy, shared sign, lines meet at the means, r = ±1 means the lines coincide).
• Then move on to probability, which uses the same careful counting.