CA Foundation · Quantitative Aptitude

Measures of Central Tendency and Dispersion: formula sheet

Full chapter guide

Key formulas

Simple AM (ungrouped): x̄ = Σx ÷ n; Equivalent to Σx = n × x̄. Use this to move between mean and total.
AM of a frequency distribution: x̄ = Σfx ÷ Σf; For class intervals, x is the mid-point of the class.
Weighted AM: x̄w = Σwx ÷ Σw; w is the weight of each value x.
Combined AM of two groups: x̄ = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂); Extends to more groups by adding terms in numerator and denominator.
Step-deviation method: d = (x − A) ÷ h; x̄ = A + h × (Σfd ÷ Σf); A is an assumed mean and h is a common factor of the class width or differences.
Corrected mean: Correct Σx = Wrong Σx − wrong value + correct value; Wrong Σx = n × wrong mean. Divide the correct total by n (or the new n if items are added or removed).
Properties of AM: Σ(x − x̄) = 0; if y = a + bx then ȳ = a + b·x̄; Adding, subtracting, multiplying or dividing every item changes the mean in the same way. The mean is affected by extreme values.
Median, ungrouped data: Median = value of the (n + 1) ÷ 2 th item in sorted data; If n is even, the position ends in .5. Take the average of the two middle items.
Quartiles, ungrouped data: Qi = value of the i(n + 1) ÷ 4 th item, i = 1, 2, 3; If the position is fractional, take the lower item plus that fraction of the gap to the next item.
Deciles and percentiles, ungrouped data: Di = value of the i(n + 1) ÷ 10 th item; Pi = value of the i(n + 1) ÷ 100 th item; Sort the data first. Use the same interpolation for fractional positions.
Median, grouped data: Median = L + [(N ÷ 2 − cf) ÷ f] × h; L = lower boundary of median class, N = total frequency, cf = cumulative frequency of the class before it, f = frequency of the median class, h = class width. Classes must be continuous.
Quartile, decile, percentile, grouped data: Qi = L + [(iN ÷ 4 − cf) ÷ f] × h; Di = L + [(iN ÷ 10 − cf) ÷ f] × h; Pi = L + [(iN ÷ 100 − cf) ÷ f] × h; Same formula as the median. Only the target position changes. L, cf, f and h refer to the class that contains that position.
Equivalences: Q2 = D5 = P50 = Median; Q1 = P25; Q3 = P75; Di = P(10i); Use these to convert one question into another type.
Mode for ungrouped data: Mode = value with the highest frequency; There can be no mode, one mode or several modes.
Mode for grouped data: Mode = L + (f1 − f0) ÷ (2f1 − f0 − f2) × h; L = lower limit of modal class, f1 = frequency of modal class, f0 = frequency of class before it, f2 = frequency of class after it, h = class width. Classes must be continuous (exclusive) with equal width.
Empirical relation: Mode = 3 Median − 2 Mean; Approximate, for moderately skewed distributions.
Empirical relation (alternate form): Mean − Mode = 3 (Mean − Median); Same relation rearranged. Median = (Mode + 2 Mean) ÷ 3 and Mean = (3 Median − Mode) ÷ 2.
Inclusive to exclusive classes: Adjustment = (next lower limit − previous upper limit) ÷ 2; Subtract it from every lower limit and add it to every upper limit. For 10-19, 20-29 the adjustment is 0.5, so the classes become 9.5-19.5, 19.5-29.5.
Variance (ungrouped): σ² = Σ(x − x̄)² ÷ N; N is the number of observations. CA Foundation questions normally divide by N.
Shortcut form of variance: σ² = Σx² ÷ N − (x̄)²; Use this when the mean is not a whole number. It avoids many subtractions.
Standard deviation: σ = √variance; Always the positive root.
Grouped data (frequency distribution): σ² = Σf(x − x̄)² ÷ N = Σfx² ÷ N − (Σfx ÷ N)²; N = Σf. For classes, x is the class mark (mid-value).
Step deviation method: d = (x − A) ÷ h; σ = h × √[Σfd² ÷ N − (Σfd ÷ N)²]; A is the assumed mean and h is the common class width. Multiply by h at the end for SD, and by h² for variance.
Change of origin and scale: If y = a + bx, then σy = |b| × σx and variance of y = b² × variance of x; a (the origin shift) has no effect. A negative b does not make SD negative.
Combined mean: x̄₁₂ = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂); Find this first for the combined SD.
Combined standard deviation: σ₁₂ = √[(n₁σ₁² + n₂σ₂² + n₁d₁² + n₂d₂²) ÷ (n₁ + n₂)], where d₁ = x̄₁ − x̄₁₂ and d₂ = x̄₂ − x̄₁₂; Works for two groups. If the two means are equal, d₁ = d₂ = 0.
First n natural numbers: Variance of 1, 2, 3, …, n = (n² − 1) ÷ 12; The same holds for any n equally spaced values with common difference 1.
Coefficient of variation: CV = (σ ÷ x̄) × 100; Used to compare the spread of two series. A lower CV means more consistency.
Coefficient of variation: CV = (σ ÷ x̄) × 100; σ is the standard deviation, x̄ is the arithmetic mean. The result is a percentage. Use the mean as a positive value.
Variance to SD: σ = √Variance; If a question gives variance, take the square root before computing CV.
Consistency rule: Smaller CV ⇒ more consistent; larger CV ⇒ more variable; Compare CVs of the two series only. Do not compare SDs when means differ.
Finding SD from CV: σ = CV × x̄ ÷ 100; Use this when CV and mean are given and SD is asked.
Effect of change of origin and scale: For y = a + bx: ȳ = a + b·x̄ and σy = |b|·σx; Adding a constant does not change SD, but it changes the mean, so CV changes.
Karl Pearson's coefficient (with mode): Sk = (Mean − Mode) ÷ SD; Use when the mode is given or well defined. SD is the standard deviation, not the variance.
Karl Pearson's coefficient (with median): Sk = 3 × (Mean − Median) ÷ SD; Use when the mode is ill-defined or not given. It follows from Mode = 3 Median − 2 Mean.
Empirical relation among averages: Mode = 3 Median − 2 Mean; Holds approximately for moderately skewed distributions. It is not an exact law.
Bowley's coefficient: Sk = (Q3 + Q1 − 2 × Median) ÷ (Q3 − Q1); Always lies between −1 and +1. It depends only on the middle 50% of the data.
Order of averages: Positive skew: Mean > Median > Mode. Negative skew: Mean < Median < Mode. Symmetric: Mean = Median = Mode; Use this to check the sign of your answer.
Reading the sign: Sk > 0: positive skew. Sk < 0: negative skew. Sk = 0: symmetric; The sign is decided by the numerator in both coefficients.

Quick revision

Mean of grouped data: Σfx ÷ Σf; with step deviation, mean = A + h × (Σfd ÷ Σf).
Median lies at the (N ÷ 2)th item in a continuous distribution; use cumulative frequency to find the class.
Mode (grouped) = L + [(f1 − f0) ÷ (2f1 − f0 − f2)] × h, with f1 the modal class frequency.
Empirical relation for moderately skewed data: Mode ≈ 3 Median − 2 Mean.
For positive values that are not all equal: AM > GM > HM; they are equal only when all values are equal.
For two positive values: GM² = AM × HM.
Use GM for growth rates and ratios; use HM for averaging rates like speed over equal distances.
Range = Largest − Smallest; Quartile Deviation = (Q3 − Q1) ÷ 2.
Variance = (SD)²; SD = √[Σfx² ÷ Σf − (mean)²].
Changing origin does not change SD or range; multiplying by k multiplies SD and range by |k|.
CV = (SD ÷ Mean) × 100; the lower the CV, the more consistent the data.
Karl Pearson's skewness = (Mean − Mode) ÷ SD; positive means a longer right tail.

Common mistakes

Averaging the group means directly in a combined mean problem. Fix: Always weight by group size: (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂).
Using class limits or class widths instead of mid-points in grouped data. Fix: Write a separate x column with mid-point = (lower + upper) ÷ 2 before doing anything else.
Not sorting ungrouped data before finding the median or a quartile. Fix: Always rewrite the data in ascending order first. Do it even if the data looks nearly sorted.
Using (n + 1) in the grouped formula, or using N ÷ 2 as a position in ungrouped data. Fix: Ungrouped data uses i(n + 1) ÷ parts. Grouped data uses iN ÷ parts inside the interpolation formula.
Using the highest frequency number as the mode of a grouped table Fix: The highest frequency only identifies the modal class. Then apply the formula to get the mode.
Not converting inclusive classes to exclusive classes Fix: Subtract 0.5 from lower limits and add 0.5 to upper limits first. Then L is 29.5, not 30, for the class 30-39.
Giving the variance when SD is asked, or the reverse. Fix: Circle the word variance or SD in the question. Take the square root at the end only when SD is asked.
Multiplying by h instead of h² for variance in the step deviation method. Fix: SD scales by h. Variance scales by h². Decide which you have before multiplying.
Choosing the series with the larger CV as more consistent. Fix: Remember that CV measures variation. Less variation means more consistency, so pick the smaller CV.
Comparing standard deviations instead of CVs when the means differ. Fix: Whenever the means are different, compute CV. Only use SD alone if the means and units are the same.

Exam tips

Convert every mean into a total first. Most questions then become one line of arithmetic.
In combined mean questions, check that your answer lies between the group means and nearer the larger group before marking it.
Expect corrected-mean questions with two errors, or with an item added or removed. Update both the total and the count.
For grouped data, a smart choice of A and h can cut your calculation time. Always recheck the sign of d for classes below A.
Avoid guessing when you cannot eliminate at least two options, because each wrong answer costs 0.25 marks.
Look at the options before calculating. In grouped data, finding the class that holds the target position usually removes two or three options.
Read the question for the word used: quartile, decile or percentile. Write the target position (such as 3N ÷ 4 or 8N ÷ 10) at the top before touching the table.
Check whether the classes are inclusive or continuous. Questions sometimes use inclusive classes to test boundary correction.