Averages, Spread & Cumulative Frequency

Averages describe the centre of a dataset, while spread describes how far the values are from each other. Two classes can have the same mean score but very different consistency, so both centre and spread are needed to understand the data properly.

CSEC questions often move from calculation to interpretation: find the mean, median, range, quartiles, or cumulative frequency, then say what the result tells you. Do not treat these as isolated formulas. After calculating, write a short sentence explaining what the number means in the context of the question.

A measure of central tendency represents the "typical" or "average" value in a dataset.

Mean (Average)

The mean uses every value, so it is affected by very large or very small outliers. It is useful when the data is fairly balanced.

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

For raw data:

Example

Scores: 5, 7, 8, 9, 5

$\text{Mean} = \frac{5 + 7 + 8 + 9 + 5}{5} = \frac{34}{5} = 6.8$

For ungrouped frequency data:

$\text{Mean} = \frac{\sum(value \times frequency)}{\sum frequency}$

Example

Score	Frequency
5	2
7	1
8	1
9	1

$\text{Mean} = \frac{(5 \times 2) + (7 \times 1) + (8 \times 1) + (9 \times 1)}{2+1+1+1} = \frac{10+7+8+9}{5} = \frac{34}{5} = 6.8$

For grouped data:

Use class midpoints:

$\text{Mean} = \frac{\sum(\text{midpoint} \times \text{frequency})}{\sum \text{frequency}}$

Example

Class	Frequency	Midpoint
10-19	3	14.5
20-29	5	24.5
30-39	2	34.5

$\text{Mean} = \frac{(14.5 \times 3) + (24.5 \times 5) + (34.5 \times 2)}{3+5+2}$ $= \frac{43.5 + 122.5 + 69}{10} = \frac{235}{10} = 23.5$

Median

The median is the middle value after ordering the data. It is useful when outliers would distort the mean.

The median is the middle value when data is arranged in order.

For raw data:

Arrange values from smallest to largest
If odd number of values: median is the middle one
If even number of values: median is the average of the two middle ones

Example

Scores: 5, 5, 7, 8, 9 (5 values, odd)

Median = 7 (the 3rd value)

Scores: 5, 5, 7, 8, 9, 10 (6 values, even)

Median = (7 + 8) ÷ 2 = 7.5

For grouped data:

Use the cumulative frequency table and interpolation:

$\text{Median} = L + \frac{\frac{n}{2} - CF}{f} \times w$

Where:

$L$ = lower boundary of median class
$n$ = total frequency
$CF$ = cumulative frequency before median class
$f$ = frequency of median class
$w$ = class width

Example

Class	Frequency	Cumulative
10-19	3	3
20-29	5	8
30-39	2	10

Total = 10, so median position = 10÷2 = 5

Median class is 20-29 (cumulative frequency reaches 5 here)

$\text{Median} = 19.5 + \frac{5 - 3}{5} \times 10 = 19.5 + \frac{2}{5} \times 10 = 19.5 + 4 = 23.5$

Number line: ordered values with median marked

Mode

The mode identifies the most common value. It is especially useful for categorical data, where mean and median may not make sense.

The mode is the value that appears most often.

Example

Scores: 5, 5, 5, 7, 8, 9, 9

Mode = 5 (appears 3 times)

Data: 2, 5, 5, 7, 7, 9

Two modes: 5 and 7 (both appear twice) = bimodal

Data: 2, 5, 7, 9

No mode (all appear once) = no mode

Choosing Mean, Median, or Mode

Choosing the average is a reasoning skill. The best measure depends on the shape of the data and what the question is trying to describe.

Use MEAN when:

Data is roughly symmetric
No extreme outliers
You want to use all values
Example: class average on a test

Use MEDIAN when:

Data has outliers or is skewed
You want the "typical" middle value
Example: house prices (skewed by luxury homes)

Use MODE when:

Categorical data (colors, preferences)
Discrete data with clear peaks
Example: favorite color, most common shoe size

Exam Tip

Example: Which average?

House prices: 120,000, 125,000, 130,000, 140,000, 2,000,000

Mean = (120+125+130+140+2000)÷5 = 502,300 (way too high!)
Median = 130,000 (better,the luxury home is an outlier)
Mode = no mode

Best answer: Median, because the data has an outlier.

Measures of Spread (Dispersion)

Spread measures how far apart the data values are from each other.

Range

Range gives a quick sense of spread, but it only uses the smallest and largest values. One unusual value can make the range misleading.

$\text{Range} = \text{Maximum value} - \text{Minimum value}$

Example

Scores: 5, 7, 8, 9, 5

Range = 9 - 5 = 4

Problem: Only uses the extreme values. Doesn't show middle spread.

Quartiles and Interquartile Range

Quartiles split ordered data into four parts. The interquartile range focuses on the middle half of the data, so it is less affected by extremes.

Quartiles divide the data into 4 equal parts.

Q₁ (1st quartile) = 25th percentile
Q₂ (2nd quartile) = 50th percentile = median
Q₃ (3rd quartile) = 75th percentile

Interquartile Range (IQR): $\text{IQR} = Q_3 - Q_1$

This shows the spread of the middle 50% of data.

Example

Test scores: 5, 6, 7, 7, 8, 8, 8, 9, 9, 10 (10 values)

Arrange in order: 5, 6, 7, 7, 8, 8, 8, 9, 9, 10

Q₁ position = (10+1) ÷ 4 = 2.75 → between 2nd and 3rd values = 6 + 0.75(7-6) = 6.75

Q₂ position = (10+1) ÷ 2 = 5.5 → between 5th and 6th values = 8

Q₃ position = 3(10+1) ÷ 4 = 8.25 → between 8th and 9th values = 9 + 0.25(9-9) = 9

$\text{IQR} = Q_3 - Q_1 = 9 - 6.75 = 2.25$

Box-and-whisker plot showing min, Q₁, median, Q₃, max

Semi-Interquartile Range

The semi-interquartile range is half of the IQR. It gives a compact measure of spread around the middle of the dataset.

$\text{Semi-IQR} = \frac{\text{IQR}}{2} = \frac{Q_3 - Q_1}{2}$

Example

Using the example above:

$\text{Semi-IQR} = \frac{2.25}{2} = 1.125$

Remember

Range: Uses only extremes, sensitive to outliers
IQR: Shows spread of middle 50%, ignores outliers
Semi-IQR: Half of IQR, useful for comparison

Cumulative Frequency and Ogives

Cumulative Frequency Table

Cumulative frequency is a running total. It answers questions like "how many values are less than or equal to this point?"

Cumulative frequency = total count up to and including that class.

Example

Class	Frequency	Cumulative Frequency
10-19	3	3
20-29	5	3+5 = 8
30-39	7	8+7 = 15
40-49	4	15+4 = 19
50-59	1	19+1 = 20

Cumulative Frequency Curve (Ogive)

An ogive turns cumulative totals into a graph. It is useful for estimating medians, quartiles, and percentiles from grouped data.

An ogive is an S-shaped curve showing cumulative frequency.

How to draw:

Use class boundaries on x-axis (not class limits)
Use cumulative frequency on y-axis
Plot point at upper boundary of each class
Connect points with a smooth curve

Example

Using the table above:

Upper Boundary	Cumulative Frequency
19.5	3
29.5	8
39.5	15
49.5	19
59.5	20

Ogive (cumulative frequency curve) — S-shaped, plotted at upper class boundaries

Reading from an Ogive

To read from an ogive, move horizontally from the cumulative frequency value to the curve, then down to the data value. This is an estimate, so use the graph carefully.

You can read:

Quartiles: Q₁ at 25% of total, Q₂ at 50%, Q₃ at 75%
Percentiles: any value's percentage position
Median: where cumulative frequency = n/2
Frequencies above/below a given value

Example

From the ogive above (n=20):

Q₁ (25% of 20 = 5): Read across from cumulative frequency 5 to curve, then down to x-axis ≈ 22

Median (50% of 20 = 10): Read from cumulative 10 ≈ 32

Q₃ (75% of 20 = 15): Read from cumulative 15 = 39.5

Ogive with quartile lines: Q₁ (cf=5), Q₂ (cf=10), Q₃ (cf=15)

A measure of central tendency represents the "typical" or "average" value in a dataset.

Mean (Average)

The mean uses every value, so it is affected by very large or very small outliers. It is useful when the data is fairly balanced.

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

For raw data:

Example

Scores: 5, 7, 8, 9, 5

$\text{Mean} = \frac{5 + 7 + 8 + 9 + 5}{5} = \frac{34}{5} = 6.8$

For ungrouped frequency data:

$\text{Mean} = \frac{\sum(value \times frequency)}{\sum frequency}$

Example

Score	Frequency
5	2
7	1
8	1
9	1

$\text{Mean} = \frac{(5 \times 2) + (7 \times 1) + (8 \times 1) + (9 \times 1)}{2+1+1+1} = \frac{10+7+8+9}{5} = \frac{34}{5} = 6.8$

For grouped data:

Use class midpoints:

$\text{Mean} = \frac{\sum(\text{midpoint} \times \text{frequency})}{\sum \text{frequency}}$

Example

Class	Frequency	Midpoint
10-19	3	14.5
20-29	5	24.5
30-39	2	34.5

$\text{Mean} = \frac{(14.5 \times 3) + (24.5 \times 5) + (34.5 \times 2)}{3+5+2}$ $= \frac{43.5 + 122.5 + 69}{10} = \frac{235}{10} = 23.5$

Median

The median is the middle value after ordering the data. It is useful when outliers would distort the mean.

The median is the middle value when data is arranged in order.

For raw data:

Arrange values from smallest to largest
If odd number of values: median is the middle one
If even number of values: median is the average of the two middle ones

Example

Scores: 5, 5, 7, 8, 9 (5 values, odd)

Median = 7 (the 3rd value)

Scores: 5, 5, 7, 8, 9, 10 (6 values, even)

Median = (7 + 8) ÷ 2 = 7.5

For grouped data:

Use the cumulative frequency table and interpolation:

$\text{Median} = L + \frac{\frac{n}{2} - CF}{f} \times w$

Where:

$L$ = lower boundary of median class
$n$ = total frequency
$CF$ = cumulative frequency before median class
$f$ = frequency of median class
$w$ = class width

Example

Class	Frequency	Cumulative
10-19	3	3
20-29	5	8
30-39	2	10

Total = 10, so median position = 10÷2 = 5

Median class is 20-29 (cumulative frequency reaches 5 here)

$\text{Median} = 19.5 + \frac{5 - 3}{5} \times 10 = 19.5 + \frac{2}{5} \times 10 = 19.5 + 4 = 23.5$

Number line: ordered values with median marked

Mode

The mode identifies the most common value. It is especially useful for categorical data, where mean and median may not make sense.

The mode is the value that appears most often.

Example

Scores: 5, 5, 5, 7, 8, 9, 9

Mode = 5 (appears 3 times)

Data: 2, 5, 5, 7, 7, 9

Two modes: 5 and 7 (both appear twice) = bimodal

Data: 2, 5, 7, 9

No mode (all appear once) = no mode

Choosing Mean, Median, or Mode

Choosing the average is a reasoning skill. The best measure depends on the shape of the data and what the question is trying to describe.

Use MEAN when:

Data is roughly symmetric
No extreme outliers
You want to use all values
Example: class average on a test

Use MEDIAN when:

Data has outliers or is skewed
You want the "typical" middle value
Example: house prices (skewed by luxury homes)

Use MODE when:

Categorical data (colors, preferences)
Discrete data with clear peaks
Example: favorite color, most common shoe size

Exam Tip

Example: Which average?

House prices: 120,000, 125,000, 130,000, 140,000, 2,000,000

Mean = (120+125+130+140+2000)÷5 = 502,300 (way too high!)
Median = 130,000 (better,the luxury home is an outlier)
Mode = no mode

Best answer: Median, because the data has an outlier.

Measures of Spread (Dispersion)

Spread measures how far apart the data values are from each other.

Range

Range gives a quick sense of spread, but it only uses the smallest and largest values. One unusual value can make the range misleading.

$\text{Range} = \text{Maximum value} - \text{Minimum value}$

Example

Scores: 5, 7, 8, 9, 5

Range = 9 - 5 = 4

Problem: Only uses the extreme values. Doesn't show middle spread.

Quartiles and Interquartile Range

Quartiles split ordered data into four parts. The interquartile range focuses on the middle half of the data, so it is less affected by extremes.

Quartiles divide the data into 4 equal parts.

Q₁ (1st quartile) = 25th percentile
Q₂ (2nd quartile) = 50th percentile = median
Q₃ (3rd quartile) = 75th percentile

Interquartile Range (IQR): $\text{IQR} = Q_3 - Q_1$

This shows the spread of the middle 50% of data.

Example

Test scores: 5, 6, 7, 7, 8, 8, 8, 9, 9, 10 (10 values)

Arrange in order: 5, 6, 7, 7, 8, 8, 8, 9, 9, 10

Q₁ position = (10+1) ÷ 4 = 2.75 → between 2nd and 3rd values = 6 + 0.75(7-6) = 6.75

Q₂ position = (10+1) ÷ 2 = 5.5 → between 5th and 6th values = 8

Q₃ position = 3(10+1) ÷ 4 = 8.25 → between 8th and 9th values = 9 + 0.25(9-9) = 9

$\text{IQR} = Q_3 - Q_1 = 9 - 6.75 = 2.25$

Box-and-whisker plot showing min, Q₁, median, Q₃, max

Semi-Interquartile Range

The semi-interquartile range is half of the IQR. It gives a compact measure of spread around the middle of the dataset.

$\text{Semi-IQR} = \frac{\text{IQR}}{2} = \frac{Q_3 - Q_1}{2}$

Example

Using the example above:

$\text{Semi-IQR} = \frac{2.25}{2} = 1.125$

Remember

Range: Uses only extremes, sensitive to outliers
IQR: Shows spread of middle 50%, ignores outliers
Semi-IQR: Half of IQR, useful for comparison

Cumulative Frequency and Ogives

Cumulative Frequency Table

Cumulative frequency is a running total. It answers questions like "how many values are less than or equal to this point?"

Cumulative frequency = total count up to and including that class.

Example

Class	Frequency	Cumulative Frequency
10-19	3	3
20-29	5	3+5 = 8
30-39	7	8+7 = 15
40-49	4	15+4 = 19
50-59	1	19+1 = 20

Cumulative Frequency Curve (Ogive)

An ogive turns cumulative totals into a graph. It is useful for estimating medians, quartiles, and percentiles from grouped data.

An ogive is an S-shaped curve showing cumulative frequency.

How to draw:

Use class boundaries on x-axis (not class limits)
Use cumulative frequency on y-axis
Plot point at upper boundary of each class
Connect points with a smooth curve

Example

Using the table above:

Upper Boundary	Cumulative Frequency
19.5	3
29.5	8
39.5	15
49.5	19
59.5	20

Ogive (cumulative frequency curve) — S-shaped, plotted at upper class boundaries

Reading from an Ogive

To read from an ogive, move horizontally from the cumulative frequency value to the curve, then down to the data value. This is an estimate, so use the graph carefully.

You can read:

Quartiles: Q₁ at 25% of total, Q₂ at 50%, Q₃ at 75%
Percentiles: any value's percentage position
Median: where cumulative frequency = n/2
Frequencies above/below a given value

Example

From the ogive above (n=20):

Q₁ (25% of 20 = 5): Read across from cumulative frequency 5 to curve, then down to x-axis ≈ 22

Median (50% of 20 = 10): Read from cumulative 10 ≈ 32

Q₃ (75% of 20 = 15): Read from cumulative 15 = 39.5

Ogive with quartile lines: Q₁ (cf=5), Q₂ (cf=10), Q₃ (cf=15)