Measures of Central Tendency and Location

Mean, median, mode, quartiles, percentiles, interpolation and coding

SPEC 2.3 • DATA PRESENTATION

Three Types of Average

Mean (= The Average)

What: Sum of values ÷ count

Use: Numerical data, no outliers

Pro: Uses all values

Con: Affected by extremes

Median

What: Middle value when ordered

Use: Skewed data, outliers present

Pro: Robust to extremes

Con: Ignores most values

Mode

What: Most frequent value

Use: Categorical data

Pro: Works for any data type

Con: May not be unique

Key Insight

For symmetric distributions, mean ≈ median ≈ mode. For skewed data, they differ significantly.

Mean Formulas

Two Cases

Raw data:

\overset{x}{ˉ} = \frac{\sum x}{n}

Frequency table:

\overset{x}{ˉ} = \frac{\sum f x}{\sum f}

Raw Data

Add all values, divide by count

Example: 3, 5, 7, 9

Mean = (3+5+7+9)/4 = 6

Frequency Table

Multiply each x by its frequency f

Sum the products, divide by Σf

Use a table to organise working

01Mean from Raw Data

Test scores for 8 students:

72, 85, 68, 91, 77, 84, 79, 88

Calculate the mean score.

Sum: 72 + 85 + 68 + 91 + 77 + 84 + 79 + 88 = 644

Count: n = 8

\overset{x}{ˉ} = \frac{644}{8} = 80.5

marks

02Mean from Frequency Table

Find the mean number of pets per household:

Pets (x)Households (f)

08

112

215

310

45

\overset{x}{ˉ} = \frac{\sum f x}{\sum f} = \frac{92}{50} = 1.84

pets per household

Median and Mode

Median

The middle value when data is ordered.

Position = \frac{n + 1}{2}

Odd n: Take that position directly

Even n: Mean of the two middle values

Mode

The most frequent value.

Unimodal: One mode

Bimodal: Two modes

No mode: All values appear equally

Important

Data must be ordered before finding the median! Always arrange in ascending order first.

03Finding Median and Mode

Times (minutes) for 9 students to complete a puzzle:

12, 15, 18, 15, 22, 15, 19, 17, 21

Find the median and mode.

First, order the data: 12, 15, 15, 15, 17, 18, 19, 21, 22

Median: n = 9, Position = (9+1)/2 = 5th value

The 5th value is 17 minutes

Mode: 15 appears 3 times (most frequent)

Mode = 15 minutes

Quartiles (Discrete Data)

Position Formulas

Q1:

\frac{n + 1}{4}

Q2:

\frac{n + 1}{2}

Q3:

\frac{3 ( n + 1 )}{4}

Round Up Rule

For discrete data, if the position is not a whole number, round UP to get the position.

Position = 3.25 → take the 4th value

Position = 6.75 → take the 7th value

Interquartile Range (IQR)

IQR = Q3 − Q1

Measures the spread of the middle 50% of data.

04Quartiles (Discrete)

Find Q1, Q2, Q3 and IQR for this data (n = 10):

14, 7, 22, 3, 16, 8, 25, 5, 19, 12

First, order the data: 3, 5, 7, 8, 12, 14, 16, 19, 22, 25

Q1 position: (10+1)/4 = 2.75 → round up → 3rd value

Q1 = 7

Q2 position: (10+1)/2 = 5.5 → round up → 6th value

Q2 = 14

Q3 position: 3(10+1)/4 = 8.25 → round up → 9th value

Q3 = 22

IQR = Q3 − Q1 = 22 − 7 = 15

Mean with Midpoints

For grouped data, we don't know exact values. Use the class midpoint as an estimate.

Formula

\overset{x}{ˉ} = \frac{\sum f m}{\sum f}

where m = midpoint of each class

Finding Midpoints

Midpoint = (lower + upper) ÷ 2

Class 10 ≤ x < 20:

midpoint = (10+20)/2 = 15

Modal Class

The class with the highest frequency.

This is the grouped equivalent of the mode - we can't find an exact mode, only identify which class contains most values.

Important

The mean gives an estimate only, since we assume values are spread evenly within each class.

05Mean from Grouped Data

Find the estimated mean time:

Time (mins)Frequency

0 ≤ t < 105

10 ≤ t < 2012

20 ≤ t < 3018

30 ≤ t < 4010

\overset{x}{ˉ} = \frac{\sum f m}{\sum f} = \frac{1005}{45} = 22.3

minutes (estimate)

Linear Interpolation

For grouped data, we estimate values using proportion. The position lies the same fraction through the Cumulative Frequency (CF) as the value lies through the class.

Class boundaries

Cumulative

Frequency

What is m?

M = Median Q₁ = Lower Quartile Q₃ = Upper Quartile Pₙ = nth Percentile

Proportion Method

\frac{m - L B}{U B - L B} = \frac{p - F}{f}

Rearranges to:

m = L B + \frac{( p - F )}{f} \times w

Key

LB = Lower Boundary UB = Upper Boundary w = class width p = position

F = Cumulative Frequency before class f = frequency of this class

06Using Interpolation

Use interpolation to estimate the median:

Time (s)fCF

0 ≤ t < 1088

10 ≤ t < 201523

20 ≤ t < 302548

30 ≤ t < 401260

Median position:

\frac{n}{2} = \frac{60}{2} =

30th value

The 30th value is in class 20 ≤ t < 30 (CF goes from 23 to 48)

Using interpolation:

\frac{M - 20}{30 - 20} = \frac{30 - 23}{48 - 23}

\frac{M - 20}{10} = \frac{7}{25}

M - 20 = 2.8

Median = 22.8 seconds

Percentiles

Definition

The p-th percentile is the value below which p% of the data falls.

P_{p} position = \frac{p \times n}{100}

Common Percentiles

P₂₅ = Q1

P₅₀ = Q2 = Median

P₇₅ = Q3

P₁₀, P₉₀ = Deciles

Finding Percentiles

Discrete: Round up position

Grouped: Use interpolation

Same methods as quartiles!

07Percentiles

Internet usage time (mins) for 70 students. Find estimates for (a) the 10th percentile and (b) the 90th percentile.

Time (mins)30–3132–3334–3637–39

Frequency2253013

CF2275770

(a) P₁₀ position:

\frac{10 \times 70}{100} =

7th value → class 32–33 (boundaries: 31.5–33.5)

\frac{P _{10} - 31.5}{33.5 - 31.5} = \frac{7 - 2}{27 - 2} \Rightarrow \frac{P _{10} - 31.5}{2} = \frac{5}{25} \Rightarrow

P₁₀ = 31.9 mins

(b) P₉₀ position:

\frac{90 \times 70}{100} =

63rd value → class 37–39 (boundaries: 36.5–39.5)

\frac{P _{90} - 36.5}{39.5 - 36.5} = \frac{63 - 57}{70 - 57} \Rightarrow \frac{P _{90} - 36.5}{3} = \frac{6}{13} \Rightarrow

P₉₀ = 37.9 mins

What is Coding?

Concept

Coding transforms data to make calculations easier. Common form:

y = \frac{x - a}{b}

where a and b are chosen constants

Why Use Coding?

Simplifies large or awkward numbers

Makes mental arithmetic easier

Reduces calculation errors

Often used in exam questions

Example

Data: 1005, 1010, 1015, 1020 → Code with y = (x − 1000)/5 → gives: 1, 2, 3, 4

Reversing the Code

If

y = \frac{x - a}{b}

, we can find the original mean from the coded mean:

Key Formula

\overset{x}{ˉ} = b \overset{y}{ˉ} + a

Method

Calculate ȳ from the coded data

Identify a and b from the coding formula

Apply: x̄ = bȳ + a

Common Error

In y = (x − a)/b: the value subtracted is a, the value divided by is b.

08Using Coding

The heights of 5 plants are coded using y = (x − 50)/10

The coded data has ȳ = 2.4

Find the mean of the original heights.

Identify the coding: y = (x − 50)/10

So a = 50, b = 10

Using x̄ = bȳ + a:

x̄ = 10(2.4) + 50 = 24 + 50 = 74 cm

Practice

Journey times for 60 commuters:

Time (mins)fCF

0 ≤ t < 151010

15 ≤ t < 302434

30 ≤ t < 451852

45 ≤ t < 60860

Use interpolation to find: (a) median (b) Q1 (c) Q3

(a) Median: Position = 60/2 = 30th value → class 15-30

\frac{M - 15}{30 - 15} = \frac{30 - 10}{34 - 10} \Rightarrow \frac{M - 15}{15} = \frac{20}{24} \Rightarrow

M = 27.5 mins

(b) Q1: Position = 60/4 = 15th value → class 15-30

\frac{Q _{1} - 15}{15} = \frac{15 - 10}{24} \Rightarrow

Q1 = 18.1 mins

(c) Q3: Position = 3×60/4 = 45th value → class 30-45

\frac{Q _{3} - 30}{15} = \frac{45 - 34}{18} \Rightarrow

Q3 = 39.2 mins

Practice

Data is coded using y = (x − 200)/5

The coded data has: Σy = 36 and n = 9

Find the mean of the original data.

Step 1: Find coded mean

ȳ = Σy/n = 36/9 = 4

Step 2: Identify a and b

From y = (x − 200)/5: a = 200, b = 5

Step 3: Find original mean

x̄ = bȳ + a = 5(4) + 200 = 220

Summary

Key Formulas

MeasureDiscreteGrouped

Mean

\frac{\sum f x}{\sum f}

\frac{\sum f m}{\sum f}

(midpoints)

Median position

\frac{n + 1}{2}

(round up)

\frac{n}{2}

(interpolate)

Q1, Q3Round up positionInterpolate

Interpolation—

L + \frac{p - F}{f} \times w

Coding

\overset{x}{ˉ} = b \overset{y}{ˉ} + a

Key Exam Skills

Choose appropriate method for discrete vs grouped data

Round UP for discrete quartile positions

Use interpolation for grouped data