Measures of central tendency

Overview

Measures of central tendency describe the centre of a set of data. The three main measures are mean, median, and mode. They answer slightly different questions:

• The mean asks, "What is the balanced average?"
• The median asks, "What is the middle value?"
• The mode asks, "What value or class occurs most often?"

These measures help learners summarize many values using one representative value. But a representative value must be chosen carefully. A very high or very low value can pull the mean away from the middle, while the median may better describe the centre of an uneven data set. The mode is useful when the most common value or class matters.

In Form IV statistics, central tendency often appears after Frequency distribution. Learners are expected not only to calculate, but also to read grouped tables, identify the correct class, and give rounded answers when required.

Prerequisites

• Ordering numbers from smallest to largest.
• Addition, multiplication, division, and substitution into formulas.
• Frequency distribution for class intervals, frequencies, midpoints, and cumulative frequency.
• Approximations, rounding, significant figures, and decimal places for answers requested to decimal places or significant figures.
• Basic algebraic rearrangement for assumed-mean work.
• Careful table reading, especially when a question gives grouped or cumulative frequencies.

Learning Scope

This chapter covers mean, median, and mode for ungrouped data, frequency tables, and grouped data. It includes class marks, total frequency, cumulative frequency, median class, modal class, grouped-data formulas, assumed mean, interpretation, and common checking routines.

It does not fully teach the construction of histograms or ogives, though those graphs can be used to estimate mode and median. Graphical treatment belongs mainly to Histogram, frequency polygon, and cumulative frequency curve.

Subtopics

Mean

The mean is the sum of all values divided by the number of values.

$$ \text{mean}=\frac{\text{sum of values}}{\text{number of values}} $$

For the values $4, 6, 7, 9$:

$$ \text{mean}=\frac{4+6+7+9}{4}=\frac{26}{4}=6.5 $$

The mean uses every value, so it is sensitive to unusually large or small values.

Mean From A Frequency Table

When values have frequencies, multiply each value by its frequency.

$$ \bar{x}=\frac{\sum fx}{\sum f} $$

Here, $x$ is the value and $f$ is its frequency.

| Score $x$ | Frequency $f$ | $fx$ | | ---: | ---: | ---: | | 2 | 3 | 6 | | 3 | 5 | 15 | | 4 | 2 | 8 | | Total | 10 | 29 |

Therefore:

$$ \bar{x}=\frac{29}{10}=2.9 $$

The total frequency $\sum f$ is the number of observations.

Mean From Grouped Data

For grouped data, the exact values inside each class are not known. The class mark or midpoint is used to represent the class.

$$ \bar{x}=\frac{\sum fm}{\sum f} $$

where $m$ is the class mark.

This gives an estimated mean because grouped data have already compressed the original values.

Assumed Mean Method

The assumed mean method can make grouped-data mean calculations shorter when class marks are large.

Use:

$$ \bar{x}=A+\frac{\sum fd}{\sum f} $$

where:

• $A$ is the assumed mean.
• $d=m-A$.
• $m$ is the class mark.

If class widths are equal, a coded version may use:

$$ \bar{x}=A+\frac{\sum fu}{\sum f}\times c $$

where $u=\frac{m-A}{c}$ and $c$ is the class width.

The method is not a new average. It is a shorter route to the same grouped mean.

Median

The median is the middle value after the data have been arranged in order.

For an odd number of values, the median is the middle value.

For an even number of values, the median is the mean of the two middle values.

Example:

$$ 3,\ 5,\ 8,\ 11,\ 13 $$

The median is $8$.

For:

$$ 3,\ 5,\ 8,\ 11 $$

the median is:

$$ \frac{5+8}{2}=6.5 $$

Median From A Frequency Table

Use cumulative frequency to locate the middle position.

If the total frequency is $N$:

• For odd $N$, locate the $\frac{N+1}{2}$th value.
• For even $N$, locate the $\frac{N}{2}$th and $\frac{N}{2}+1$th values, then average them if needed.

The cumulative frequency column tells where each value range ends.

Median From Grouped Data

For grouped data, use the median class and the formula:

$$ \text{Median}=L+\left(\frac{\frac{N}{2}-C_f}{f}\right)c $$

where:

• $L$ is the lower boundary of the median class.
• $N$ is the total frequency.
• $C_f$ is the cumulative frequency before the median class.
• $f$ is the frequency of the median class.
• $c$ is the class width.

The median class is the first class whose cumulative frequency is at least $\frac{N}{2}$.

Mode

The mode is the value that occurs most often.

For:

$$ 2,\ 3,\ 3,\ 4,\ 5,\ 5,\ 5,\ 6 $$

the mode is $5$ because it occurs $3$ times.

A data set may have no mode, one mode, or more than one mode.

Modal Class And Grouped Mode

In grouped data, the modal class is the class with the highest frequency.

To estimate the grouped mode, a common formula is:

$$ \text{Mode}=L+\left(\frac{d_1}{d_1+d_2}\right)c $$

where:

• $L$ is the lower boundary of the modal class.
• $d_1$ is the modal class frequency minus the previous class frequency.
• $d_2$ is the modal class frequency minus the next class frequency.
• $c$ is the class width.

The grouped mode is an estimate because it assumes the values within classes behave smoothly.

Choosing Mean, Median, Or Mode

Use the mean when all values should contribute to the answer and there are no extreme values that distort the result too much.

Use the median when the middle position matters or when extreme values make the mean misleading.

Use the mode when the most common value, category, or class matters.

For exam calculations, use the measure requested by the question. For interpretation, explain what that measure tells about the data.

Comparing Mean And Median

The difference between mean and median can show whether values are balanced or pulled by unusually high or low observations.

If the mean is much greater than the median, high values may be pulling the average upward. If the mean is much less than the median, low values may be pulling it downward. This is a signal, not a proof; the original data or table should still be inspected.

Key Terms

• Measure of central tendency: A value used to describe the centre of a data set.
• Mean: The sum of values divided by the number of values.
• Median: The middle value when data are arranged in order.
• Mode: The value that occurs most often.
• Frequency: The number of times a value or class occurs.
• Class mark: The midpoint of a class interval.
• Cumulative frequency: A running total of frequencies.
• Median class: The class containing the middle position.
• Modal class: The class with the highest frequency.
• Assumed mean: A convenient chosen value used to simplify mean calculation.
• Deviation: The difference between a class mark and the assumed mean.
• Outlier: A value far from most other values.

Worked Examples

Example 1: Mean, Median, And Mode For Raw Data

Find the mean, median, and mode of:

$$ 6,\ 4,\ 8,\ 4,\ 10,\ 5,\ 4 $$

Mean:

$$ \text{mean}=\frac{6+4+8+4+10+5+4}{7}=\frac{41}{7}=5.857... $$

Median: arrange the values.

$$ 4,\ 4,\ 4,\ 5,\ 6,\ 8,\ 10 $$

There are $7$ values, so the middle value is the $4$th value. The median is $5$.

Mode: $4$ occurs most often. The mode is $4$.

Final answers:

$$ \text{mean}=5.857...,\quad \text{median}=5,\quad \text{mode}=4 $$

Example 2: Mean From A Frequency Table

Find the mean.

| Score $x$ | Frequency $f$ | | ---: | ---: | | 1 | 2 | | 2 | 4 | | 3 | 3 | | 4 | 1 |

Add the $fx$ column:

| Score $x$ | Frequency $f$ | $fx$ | | ---: | ---: | ---: | | 1 | 2 | 2 | | 2 | 4 | 8 | | 3 | 3 | 9 | | 4 | 1 | 4 | | Total | 10 | 23 |

Then:

$$ \bar{x}=\frac{\sum fx}{\sum f}=\frac{23}{10}=2.3 $$

Final answer:

$$ 2.3 $$

Example 3: Mean From Grouped Data

Find the mean mark.

| Marks | Frequency | | --- | ---: | | 40-49 | 2 | | 50-59 | 4 | | 60-69 | 7 | | 70-79 | 5 | | 80-89 | 2 |

First find class marks.

| Marks | Frequency $f$ | Class mark $m$ | $fm$ | | --- | ---: | ---: | ---: | | 40-49 | 2 | 44.5 | 89.0 | | 50-59 | 4 | 54.5 | 218.0 | | 60-69 | 7 | 64.5 | 451.5 | | 70-79 | 5 | 74.5 | 372.5 | | 80-89 | 2 | 84.5 | 169.0 | | Total | 20 | | 1300.0 |

Now:

$$ \bar{x}=\frac{1300}{20}=65 $$

Final answer:

$$ 65 $$

Example 4: Median From Grouped Data

Find the median for the grouped distribution below.

| Marks | Frequency | Cumulative frequency | | --- | ---: | ---: | | 40-49 | 2 | 2 | | 50-59 | 4 | 6 | | 60-69 | 7 | 13 | | 70-79 | 9 | 22 | | 80-89 | 5 | 27 | | 90-99 | 3 | 30 |

Total frequency:

$$ N=30 $$

Middle position:

$$ \frac{N}{2}=15 $$

The first cumulative frequency at least $15$ is $22$, so the median class is $70-79$.

For this class:

$$ L=69.5,\quad C_f=13,\quad f=9,\quad c=10 $$

Substitute:

$$ \begin{aligned} \text{Median} &= 69.5+\left(\frac{15-13}{9}\right)10 \\ &= 69.5+\frac{20}{9} \\ &= 71.722... \end{aligned} $$

Final answer:

$$ 71.72 $$

to $2$ decimal places.

Example 5: Mode From Grouped Data

Estimate the mode.

| Marks | Frequency | | --- | ---: | | 50-59 | 4 | | 60-69 | 7 | | 70-79 | 12 | | 80-89 | 9 | | 90-99 | 3 |

The modal class is $70-79$ because it has the highest frequency, $12$.

Use:

$$ \text{Mode}=L+\left(\frac{d_1}{d_1+d_2}\right)c $$

For the modal class:

$$ L=69.5,\quad c=10 $$

Previous frequency is $7$, next frequency is $9$:

$$ d_1=12-7=5,\quad d_2=12-9=3 $$

Then:

$$ \begin{aligned} \text{Mode} &= 69.5+\left(\frac{5}{5+3}\right)10 \\ &= 69.5+6.25 \\ &= 75.75 \end{aligned} $$

Final answer:

$$ 75.75 $$

Example 6: Assumed Mean Method

Find the mean using assumed mean $A=64.5$.

| Marks | Frequency $f$ | Class mark $m$ | | --- | ---: | ---: | | 40-49 | 2 | 44.5 | | 50-59 | 4 | 54.5 | | 60-69 | 7 | 64.5 | | 70-79 | 5 | 74.5 | | 80-89 | 2 | 84.5 |

Compute $d=m-A$ and $fd$:

| Marks | $f$ | $m$ | $d=m-64.5$ | $fd$ | | --- | ---: | ---: | ---: | ---: | | 40-49 | 2 | 44.5 | -20 | -40 | | 50-59 | 4 | 54.5 | -10 | -40 | | 60-69 | 7 | 64.5 | 0 | 0 | | 70-79 | 5 | 74.5 | 10 | 50 | | 80-89 | 2 | 84.5 | 20 | 40 | | Total | 20 | | | 10 |

Use:

$$ \bar{x}=A+\frac{\sum fd}{\sum f} $$

So:

$$ \bar{x}=64.5+\frac{10}{20}=64.5+0.5=65 $$

Final answer:

$$ 65 $$

This matches the direct grouped mean from Example 3.

Common Mistakes

• Mistake: Finding the mean by dividing by the number of rows instead of total frequency. Correction: divide by $\sum f$.
• Mistake: Forgetting to arrange raw data before finding the median. Correction: order the data first.
• Mistake: Using class limits instead of class boundaries in grouped median or mode formulas. Correction: for $70-79$, use $69.5$ as the lower boundary.
• Mistake: Choosing the median class from frequency instead of cumulative frequency. Correction: find the class that contains $\frac{N}{2}$.
• Mistake: Calling the highest frequency the mode in grouped data. Correction: the class is the modal class; the grouped mode is an estimate from a formula.
• Mistake: Using the upper boundary instead of the lower boundary in formulas. Correction: identify $L$ before substituting.
• Mistake: Rounding too early inside a multi-step calculation. Correction: keep extra digits until the final answer.
• Mistake: Treating mean, median, and mode as always equal. Correction: they can be different, especially when data are uneven.
• Mistake: Ignoring an instruction such as "correct to 2 decimal places." Correction: round the final answer according to the question.
• Mistake: Using an assumed mean as if it must be the final mean. Correction: the assumed mean is only a calculation aid.

Practice Tasks

Foundation

1. Define mean, median, and mode.
2. Find the mean of $3, 5, 7, 9$.
3. Find the median of $8, 2, 5, 4, 7$.
4. Find the mode of $6, 4, 6, 8, 9, 6, 4$.
5. Explain why the data must be arranged before finding the median.

Skill-Building

1. Find the mean from a frequency table with values $1, 2, 3, 4$ and frequencies $3, 5, 1, 1$.
2. Add a cumulative frequency column to frequencies $4, 6, 8, 2$ and locate the median position.
3. For class $60-69$, write the class boundaries and class mark.
4. Calculate the grouped mean for classes $0-9, 10-19, 20-29$ with frequencies $2, 5, 3$.
5. Identify the modal class for frequencies $4, 9, 12, 7, 2$.

Exam-Style

1. A frequency distribution has classes $40-49, 50-59, 60-69, 70-79, 80-89, 90-99$ with frequencies $2, 4, 7, 9, 5, 3$. Find the median correct to $2$ decimal places.
2. For the same table, calculate the mean correct to $4$ significant figures.
3. A grouped table has modal class $75-79$, previous frequency $22$, modal frequency $43$, and next frequency $6$. Write the values of $L$, $d_1$, $d_2$, and $c$ for the grouped-mode formula.
4. Use an assumed mean of $77$ to calculate a grouped mean from a table supplied by your teacher.
5. Calculate the difference between the actual mean and the median of a grouped distribution, then comment on whether the two measures are close.

Challenge

1. Create a data set where the mean is greater than the median. Explain what causes the difference.
2. Create a data set with two modes and explain why it is bimodal.
3. Compare the exact mean of raw data with the estimated mean after grouping the same data. Explain why the answers may differ.
4. Explain why an ogive can help estimate the median but does not replace the grouped median calculation unless the question asks for a graphical estimate.

Generated Question Layer

• Concept questions: Ask learners to choose mean, median, or mode for different contexts and justify the choice.
• Calculation questions: Generate raw-data, frequency-table, and grouped-data questions for mean, median, and mode.
• Table-completion questions: Provide missing $fx$, class mark, cumulative frequency, $d$, or $fd$ entries.
• Error-analysis questions: Show incorrect substitutions in grouped median or mode formulas and ask learners to repair them.
• Interpretation questions: Ask learners to compare mean and median and comment on the difference.
• Rounding questions: Require final answers to specified decimal places or significant figures.

Learner Aid Opportunities

• diagram: flowchart for deciding whether to use raw-data, frequency-table, or grouped-data methods.
• chart: formula reference table for mean, median, mode, grouped median, grouped mode, and assumed mean.
• graph: link between modal class and the tallest histogram bar.
• animation: cumulative frequency building until the median class is reached.
• interactive: table calculator that asks learners to fill $fx$, class marks, cumulative frequencies, and formula variables.
• video: worked grouped median and grouped mode examples with boundary selection emphasized.
• LLM tutor: asks step-by-step prompts such as "What is $N$?", "Which class contains $N/2$?", and "What is the lower boundary?"

Exam-Derived Signals

The official 2022 examination format crosswalk maps the format group Statistics/Circles to this topic, together with Frequency distribution, Histogram, frequency polygon, and cumulative frequency curve, and Circle angle properties, theorems, tangents, chords, and radians. That format crosswalk is official. The extracted question records below are unreviewed and include some doubtful mappings, so they must not be treated as verified past-question links.

| Signal source | Status | Measures of central tendency signal | | --- | --- | --- | | exam_format_topic_crosswalk_2022.jsonl | official format crosswalk | format-041-spec-11, Statistics/Circles, $1$ item, $7.14\%$ weight, mapped to this topic and related statistics/circle topics. | | topic_frequency_2021_2025.json | unreviewed extraction | $5$ primary mapped records for topic-measures-of-central-tendency, all counted in 2021. | | question_map_2021_2025.jsonl | unreviewed extraction | Some records clearly involve mean or median, while several 2021 account-wording records look suspicious and require review. |

Recent unreviewed extracted records:

| Year | Question ID | Signal | Review note | | ---: | --- | --- | --- | | 2021 | csee_041_2021_p1_q07_a_i | Account term "Trading account" was mapped by the extractor to central tendency. | Unreviewed and likely needs subject review. | | 2021 | csee_041_2021_p1_q07_a_ii | Account term "Profit and loss account" was mapped with a ratio/proportion overlap. | Needs manual review; likely not a central tendency item. | | 2021 | csee_041_2021_p1_q07_a_iii | Account term "Balance sheet" was mapped to central tendency. | Unreviewed and likely needs subject review. | | 2021 | csee_041_2021_p1_q07_a_iv | Account term "Cash account" was mapped to central tendency. | Unreviewed and likely needs subject review. | | 2021 | csee_041_2021_p1_q11_c | Calculate difference between actual mean and median, then comment on the difference. | Unreviewed but directly relevant by wording. | | 2023 | csee_041_2023_p1_q11_a | Find median from a frequency distribution table. | Secondary topic signal; table-dependent and multi-topic. | | 2023 | csee_041_2023_p1_q11_b | Calculate mean from a frequency distribution table. | Secondary topic signal; table-dependent and multi-topic. | | 2024 | csee_041_2024_p1_q11_a_i | Find mean score using an assumed mean. | Secondary topic signal; table-dependent and mapped primarily to approximation. | | 2024 | csee_041_2024_p1_q11_a_iii | Calculate mode of grouped scores. | Secondary topic signal; table-dependent and mapped primarily to approximation. |

Source And Review Notes

• Official topic registry status: The topic identity, Form IV link, competence, source topic ID, and hub are official through data/curriculum_map.json.
• Official syllabus reference: raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf.
• Learner expansion status: The explanations, examples, and practice tasks are original learner-facing prose and remain unreviewed.
• Exam signal status: Extracted question mappings from data/question_map_2021_2025.jsonl and counts from data/topic_frequency_2021_2025.json are unreviewed. Several mappings are explicitly suspicious and should be checked before publication as past-question links.
• Format crosswalk status: data/exam_format_topic_crosswalk_2022.jsonl is an official format crosswalk and maps Statistics/Circles to this topic.
• No media assets were added. Learner aids are planning markers only.