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:

For ungrouped frequency data:

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

For grouped data:

Use class midpoints:

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

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:

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

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

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.

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

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Part 5: 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.

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

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.

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.

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

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Part 6: 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.

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:

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

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