Measures of Central Tendency and Dispersion: formula sheet

Key formulas

Requisites of a good average

Rigidly defined; easy to understand; easy to calculate; based on all observations; not unduly affected by extreme values; least affected by sampling fluctuations; suitable for further algebraic treatment

Learn this list. Questions ask which quality an average satisfies or fails.

Types of averages

Mathematical: AM, GM, HM. Positional: Median, Mode

Mean-type averages use all values. Median and mode are positional averages that depend on position or frequency.

Which averages are affected by extreme values

AM is greatly affected by extreme values. GM and HM also use all items, so they are affected too, but less than AM for large values. HM is affected most by very small values. Median and mode are not unduly affected by extreme values.

Mode resists extreme values but may be ill-defined or not unique. GM and HM are not suitable when any value is zero or negative.

Simple mean (raw data)

x̄ = Σx ÷ n

n is the number of observations.

Mean of a frequency distribution

x̄ = Σfx ÷ Σf

For classes, x is the mid-point of the class. Σf = N.

Weighted mean

x̄w = Σwx ÷ Σw

Use when items have different importance. w is the weight.

Assumed mean method

x̄ = A + Σfd ÷ Σf, where d = x − A

A is the assumed mean. Any A gives the same answer.

Step deviation method

x̄ = A + (Σfd′ ÷ Σf) × h, where d′ = (x − A) ÷ h

h is the common class width. Use only when mid-points are equally spaced.

Combined mean of two groups

x̄12 = (n1x̄1 + n2x̄2) ÷ (n1 + n2)

Weight each group mean by its size. Do not average the two means directly unless n1 = n2.

Corrected mean

Correct Σx = Wrong Σx − Wrong values + Correct values; Correct mean = Correct Σx ÷ n

First get the wrong total as n × wrong mean. If an item is omitted or added, n changes too.

Change of origin and scale

If y = a + bx, then ȳ = a + b x̄

Adding a constant to every item adds it to the mean. Multiplying every item multiplies the mean.

Median of raw data

n odd: median = value of ((n + 1) ÷ 2)th item; n even: average of the (n ÷ 2)th and (n ÷ 2 + 1)th items

Sort the data first. Never take the middle of unsorted data.

Quartiles, deciles, percentiles of raw or discrete data

Qi = i(N + 1) ÷ 4 th item; Di = i(N + 1) ÷ 10 th item; Pi = i(N + 1) ÷ 100 th item

This gives the position, not the value. For a frequency table, read the value whose cf first reaches or exceeds that position.

Median of grouped data

Median = L + [(N ÷ 2 − cf) ÷ f] × h

L = lower limit of the median class, cf = cumulative frequency before it, f = its frequency, h = class width, N = total frequency.

Quartile of grouped data

Qi = L + [(iN ÷ 4 − cf) ÷ f] × h, for i = 1, 2, 3

Locate the class containing the iN/4 th item using the cf column.

Decile of grouped data

Di = L + [(iN ÷ 10 − cf) ÷ f] × h, for i = 1 to 9

Same method; only the position changes.

Percentile of grouped data

Pi = L + [(iN ÷ 100 − cf) ÷ f] × h, for i = 1 to 99

Same method; only the position changes.

Equivalent positional values

Q2 = D5 = P50 = Median; Q1 = P25; Q3 = P75

Useful for checking and for saving time.

Mode for grouped data

Mode = L + [(f1 − f0) ÷ (2f1 − f0 − f2)] × h

L = lower limit of the modal class, f1 = frequency of the modal class, f0 = frequency of the class before it, f2 = frequency of the class after it, h = class width. Classes must be continuous and of equal width.

Equivalent form with differences

Mode = L + [Δ1 ÷ (Δ1 + Δ2)] × h

Δ1 = f1 − f0 and Δ2 = f1 − f2. Same result as the formula above.

Empirical relation

Mode = 3 Median − 2 Mean

Holds approximately for moderately skewed distributions. Not exact.

Mean from mode and median

Mean = (3 Median − Mode) ÷ 2

Rearranged form for when the mean is the missing value.

Median from mean and mode

Median = (Mode + 2 Mean) ÷ 3

Rearranged form for when the median is the missing value.

Mode of ungrouped data

Mode = value with the highest frequency

Count how often each value appears. Ties give more than one mode.

Geometric mean (ungrouped)

GM = (x₁ × x₂ × … × xₙ)^(1/n)

All values must be positive. For large numbers use logs: log GM = Σ log x ÷ n, then take the antilog.

Geometric mean (grouped data)

log GM = Σ f log x ÷ N, where N = Σ f

Here x is the mid-point of each class. Take the antilog of the result to get GM.

Weighted geometric mean

log GM = Σ w log x ÷ Σ w

Use when values carry different weights.

Harmonic mean (ungrouped)

HM = n ÷ Σ(1/x)

All values must be non-zero and positive. Do not forget the n in the numerator.

Harmonic mean (grouped data)

HM = N ÷ Σ(f/x)

x is the value or class mid-point. N = Σ f.

Weighted harmonic mean

HM = Σ w ÷ Σ(w/x)

For speeds over unequal distances, use the distances as weights.

HM of two numbers

HM = 2ab ÷ (a + b)

Shortcut for two values only, such as two speeds over equal distances.

Relation among the means

AM ≥ GM ≥ HM

For positive values. Equality holds only when all values are equal.

Relation for two numbers

GM² = AM × HM

True only for two positive numbers. Do not use it for three or more.

Average growth rate

Average growth factor = GM of the yearly factors; rate = GM − 1

A 10% rise is a factor of 1.10. A 20% fall is a factor of 0.80.

Range

Range = L − S

L = largest value, S = smallest value. For grouped data, use upper limit of the highest class minus lower limit of the lowest class.

Coefficient of range

(L − S) ÷ (L + S)

A pure number. Always uses L + S in the denominator, not L − S.

Quartile deviation

QD = (Q3 − Q1) ÷ 2

Also called semi-inter-quartile range.

Coefficient of quartile deviation

(Q3 − Q1) ÷ (Q3 + Q1)

Denominator is Q3 + Q1, not divided by 2 separately; the 2s cancel.

Mean deviation (ungrouped)

MD = Σ|x − A| ÷ n

A is the mean or median, as asked. Take absolute values.

Mean deviation (frequency data)

MD = Σf|x − A| ÷ Σf

x is the value or class mid-point.

Coefficient of mean deviation

MD ÷ A

Divide by the same average used to find MD (mean or median).

Variance (ungrouped data)

σ² = Σ(x − x̄)² ÷ n = Σx² ÷ n − (x̄)²

The second form is faster when the values are small whole numbers. Foundation questions divide by n.

Standard deviation

σ = √Variance

SD is always zero or positive. It is zero only when all values are equal.

Grouped data (direct)

σ² = Σfx² ÷ N − (Σfx ÷ N)², where N = Σf

Use x as the class mid-point for continuous classes.

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. Do not forget to multiply by h at the end.

Effect of change of origin and scale

If y = ax + b, then SD of y = |a| × SD of x, and Variance of y = a² × Variance of x

Adding or subtracting a constant does not change SD. Multiplying by a constant scales SD by |a|.

Coefficient of variation

CV = (σ ÷ x̄) × 100

Lower CV means more consistent. Higher CV means more variable.

Combined mean

x̄₁₂ = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂)

Find this first, because the combined SD needs it.

Combined standard deviation

σ₁₂ = √[(n₁σ₁² + n₂σ₂² + n₁d₁² + n₂d₂²) ÷ (n₁ + n₂)], where d₁ = x̄₁ − x̄₁₂ and d₂ = x̄₂ − x̄₁₂

The d values are measured from the combined mean, not from each other.

SD of first n natural numbers

σ = √[(n² − 1) ÷ 12]

A ready-made result for 1, 2, 3, ..., n.

Karl Pearson's coefficient of skewness

Sk = (Mean − Mode) ÷ Standard Deviation

Use when the mode is well defined. Written as (X̄ − Z) ÷ σ.

Pearson's coefficient using median

Sk = 3 × (Mean − Median) ÷ Standard Deviation

Use when the mode is not given or is ill-defined. It comes from the empirical relation Mode = 3 Median − 2 Mean.

Bowley's coefficient of skewness

Sk = (Q3 + Q1 − 2 × Median) ÷ (Q3 − Q1)

Based on quartiles only. It always lies between −1 and +1.

Empirical relation

Mode = 3 × Median − 2 × Mean

Use to find a missing mode when mean and median are given.

Order of averages

Positive skew: Mean > Median > Mode. Negative skew: Mean < Median < Mode

Zero skew: all three are equal.

Kurtosis types

Mesokurtic = normal peak; Leptokurtic = more peaked; Platykurtic = flatter

Kurtosis describes peakedness, not direction.

Quick revision

• Arithmetic mean = Σx ÷ n for raw data, and Σfx ÷ Σf for a frequency table.
• Sum of deviations of values from their arithmetic mean is zero.
• Median is the middle value of ordered data, and for grouped data it uses cumulative frequency to find the median class.
• Q1, Q2 and Q3 are the 25th, 50th and 75th percentile points, and Q2 equals the median.
• Empirical relation: Mode = 3 Median − 2 Mean, for moderately skewed data.
• Geometric mean is the nth root of the product of n positive values.
• Harmonic mean = n ÷ Σ(1/x) for positive values, and it suits averaging rates.
• For positive values that are not all equal, AM > GM > HM.
• Range = Largest − Smallest. Quartile deviation = (Q3 − Q1) ÷ 2.
• Variance = (Standard deviation)², and SD is never negative.
• Coefficient of variation = (SD ÷ Mean) × 100. A lower value means more consistency.
• Adding a constant to every value changes the mean but not the SD. Multiplying by a constant multiplies both mean and SD by that constant, using its absolute value for SD.

Common mistakes

• Saying the median is based on all observations. Fix: Median is found from the position of the middle value. It does not use the size of every value.
• Saying the arithmetic mean is not affected by extreme values. Fix: Every value enters the AM, so a very large or small value shifts it. Median is not unduly affected by this.
• Using class limits instead of mid-points in a grouped table. Fix: Always compute the mid-point first and write it as a separate column.
• Forgetting to multiply by h and add A in the step deviation method. Fix: Write the full formula A + (Σfd′ ÷ Σf) × h before substituting. Check that the answer is near A.
• Using the cumulative frequency of the median class instead of the class before it. Fix: cf in the formula is always the cf of the previous class. Underline it before substituting.
• Not sorting raw data before picking the middle. Fix: Always write the data in ascending order first, then count to the position.
• Swapping f0 and f2 in the formula Fix: f0 is the class before the modal class (lower values). f2 is the class after it. The numerator is always f1 − f0.
• Using the formula with inclusive classes such as 10–19, 20–29 Fix: Convert to exact limits first. Here the class 20–29 becomes 19.5–29.5, so L = 19.5 and h = 10.
• Using AM to average speeds over equal distances Fix: If the distance is the same and speeds differ, the right average is HM. Compute 2ab ÷ (a + b) for two speeds.
• Writing HM as the reciprocal of AM, or leaving out n Fix: Take reciprocals, add them, then divide n by that sum. HM = n ÷ Σ(1/x).

Exam tips

• Memorise the requisites list and which average fails each one.
• Questions often use phrases like 'not unduly affected by extreme values'. Link that phrase to median and mode.
• With no negative marking, always attempt every property question. Eliminate the absolute statements first.
• Remember AM is the most widely used average because it suits algebraic treatment.
• Remember mode is not unduly affected by extreme values, but it may be ill-defined or not unique.
• Watch for the word NOT in the question stem before choosing.
• Most questions are one of four types: grouped mean, combined mean, weighted mean or corrected mean. Spot the type first and pick the matching formula.
• In correction questions, work only with totals. It is faster and avoids slips.