Statistical analysis of experimental data

Overview

Physics experiments produce data. Data may be a set of readings from a ruler, stopwatch, balance, thermometer, ammeter, voltmeter, spring balance, or another measuring instrument. A single reading is useful, but repeated readings are usually better because they help a learner judge how reliable the result is.

Statistical analysis of experimental data means using simple numerical methods to describe and check readings. In Form I Physics, the most important ideas are trial, mean, range, uncertainty, variable, table, unit, and outlier. These ideas help learners decide whether data are consistent, whether a result should be repeated, and how much confidence to place in a conclusion.

Good statistical analysis does not hide experimental error. It shows the pattern in the data while also admitting the limits of the readings.

Prerequisites

• Concept of Physics - Learners should know that Physics uses observation, measurement, and evidence.
• Physical quantities and SI units - Learners should know that every measured quantity needs a number and a unit.
• Measuring instruments in Physics - Learners should be able to take careful readings and recognise range, zero error, and reading errors.
• Basic arithmetic - Learners should be able to add, subtract, divide, compare numbers, and work with decimals.
• Simple table reading - Learners should be able to read rows, columns, headings, and units.

Learning Scope

This chapter covers simple statistical treatment of experimental data in school Physics. It teaches how to record readings in a table, repeat a trial, calculate a mean, find a range, estimate simple uncertainty, notice an outlier, compare values, and write a fair conclusion from data.

This page does not teach spreadsheet processing, advanced statistics, full graph plotting, or formal error propagation. Spreadsheet work belongs to Spreadsheet processing of experimental data. Graph plotting and gradient work are developed further in Graphical presentation of experimental results and Graphs and mathematical relationships in Physics. Formula rearrangement and derived values are handled in Analytical manipulation of experimental data.

Subtopics

Data, Reading, And Trial

Data are the recorded values collected during an experiment. A reading is one measured value from an instrument. A trial is one complete attempt at taking a reading or set of readings under the same planned condition.

For example, a learner may measure the time for a trolley to move down a ramp three times:

| Trial | Time reading / s | | ---: | ---: | | 1 | $2.4$ | | 2 | $2.5$ | | 3 | $2.4$ |

The data are the three time readings. Each row is a trial.

Key insight: A trial is not just a number. It is a repeated attempt made under the same method so the learner can compare readings fairly.

Variables In A Data Table

A variable is a quantity that can change in an experiment. A clear table shows which variable was changed and which variable was measured.

Common experimental variables include:

• independent variable - the variable deliberately changed
• dependent variable - the variable measured as the result
• controlled variable - a variable kept the same to make the comparison fair

For example, in an investigation of stretching a spring, the load may be changed and the extension measured.

| Load / N | Extension trial 1 / cm | Extension trial 2 / cm | Extension trial 3 / cm | | ---: | ---: | ---: | ---: | | $1.0$ | $2.1$ | $2.0$ | $2.1$ | | $2.0$ | $4.2$ | $4.1$ | $4.2$ | | $3.0$ | $6.2$ | $6.3$ | $6.2$ |

Key insight: A table should make the variables visible. The heading should include both the quantity and the unit.

Why Repeated Readings Matter

One reading can be affected by reaction time, parallax error, a moving pointer, a rough surface, a slipping object, or a mistake in recording. Repeating readings helps the learner see whether the data are consistent.

If repeated readings are close together, the result is more trustworthy. If they are very different, the method or apparatus should be checked.

Example:

| Trial | Length / cm | | ---: | ---: | | 1 | $12.4$ | | 2 | $12.5$ | | 3 | $12.4$ |

These readings are close together.

| Trial | Length / cm | | ---: | ---: | | 1 | $12.4$ | | 2 | $15.9$ | | 3 | $12.5$ |

The second set contains one reading that is far from the others. It needs checking.

Key insight: Repetition does not make an experiment perfect, but it gives evidence about reliability.

Mean Of Repeated Readings

The mean is the sum of the readings divided by the number of readings. It is often used as the best single value when repeated readings are close.

The formula is:

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

For readings $2.4\ \text{s}$, $2.5\ \text{s}$, and $2.4\ \text{s}$:

$$ \begin{aligned} \text{mean time} &= \frac{2.4\ \text{s} + 2.5\ \text{s} + 2.4\ \text{s}}{3} \\ &= \frac{7.3\ \text{s}}{3} \\ &= 2.43\ \text{s} \end{aligned} $$

Depending on the instrument and required decimal places, this may be reported as $2.4\ \text{s}$ or $2.43\ \text{s}$.

Key insight: A mean is most useful when the readings belong to the same quantity, the same unit, and the same experimental condition.

Range Of Readings

The range shows the spread of repeated readings. It is found by subtracting the smallest reading from the largest reading.

$$ \text{range} = \text{largest reading} - \text{smallest reading} $$

For $2.4\ \text{s}$, $2.5\ \text{s}$, and $2.4\ \text{s}$:

$$ \begin{aligned} \text{range} &= 2.5\ \text{s} - 2.4\ \text{s} \\ &= 0.1\ \text{s} \end{aligned} $$

A small range usually means the readings are close together. A large range warns the learner to inspect the method, instrument, or data table.

Key insight: The mean gives a central value. The range tells how spread out the readings are.

Simple Uncertainty

Uncertainty is a statement about the possible doubt in a reading or result. At this level, uncertainty may come from the instrument scale, human reaction time, repeated readings, or changing conditions.

For a simple scale, a useful idea is that the reading is uncertain by about half the smallest division when the mark is judged by eye. If a ruler has a smallest division of $1\ \text{mm}$, a single length reading may be written with an uncertainty of about $0.5\ \text{mm}$.

For repeated readings, the range can be used to discuss uncertainty in a simple way. If repeated readings are $12.4\ \text{cm}$, $12.5\ \text{cm}$, and $12.4\ \text{cm}$, the range is $0.1\ \text{cm}$. The small range suggests low spread in the readings.

Key insight: Uncertainty does not mean the experiment failed. It is an honest statement that measured data have limits.

Outliers

An outlier is a reading that is far away from the other readings. It may be caused by a mistake, a sudden change in conditions, a poor reading, or sometimes a real but unusual event.

Example:

| Trial | Time / s | | ---: | ---: | | 1 | $3.2$ | | 2 | $3.3$ | | 3 | $8.9$ | | 4 | $3.2$ |

The reading $8.9\ \text{s}$ is an outlier because it is much larger than the others.

A learner should not simply delete an outlier without thinking. A good response is to:

• check whether the reading was copied correctly
• repeat the trial if possible
• inspect the apparatus and method
• state clearly if a reading was excluded and why

Key insight: Treat an outlier as evidence to investigate, not as a value to hide.

Tables For Statistical Analysis

A good table makes analysis easier. It should include:

• a short title or clear context
• the independent variable in the first column where possible
• repeated trial columns when readings are repeated
• a mean column when a mean is calculated
• units in column headings
• consistent decimal places where the instrument allows it

Example:

| Load / N | Extension trial 1 / cm | Extension trial 2 / cm | Extension trial 3 / cm | Mean extension / cm | | ---: | ---: | ---: | ---: | ---: | | $1.0$ | $2.1$ | $2.0$ | $2.1$ | $2.1$ | | $2.0$ | $4.2$ | $4.1$ | $4.2$ | $4.2$ | | $3.0$ | $6.2$ | $6.3$ | $6.2$ | $6.2$ |

Key insight: The unit belongs in the heading so it does not have to be repeated in every cell.

Comparing Data And Drawing A Conclusion

Statistical analysis should help answer the experimental question. A conclusion should be based on the data, not on what the learner expected before the experiment.

Weak conclusion:

"The load and extension are related."

Stronger conclusion:

"As the load increased from $1.0\ \text{N}$ to $3.0\ \text{N}$, the mean extension increased from $2.1\ \text{cm}$ to $6.2\ \text{cm}$. The data show that extension increased as load increased."

Key insight: A good conclusion names the variables, quotes data, includes units, and stays within what the readings can support.

Key Terms

• Data: Recorded values collected during an experiment.
• Reading: One measured value obtained from an instrument.
• Trial: One repeated attempt at taking a reading or set of readings under the same condition.
• Mean: The sum of readings divided by the number of readings.
• Range: The difference between the largest reading and the smallest reading.
• Uncertainty: The possible doubt or limit in a reading or result.
• Variable: A quantity that can change in an experiment.
• Table: An organised arrangement of data in rows and columns.
• Formula: A mathematical relationship or rule used to calculate a value from data.
• Unit: The standard used to express a physical quantity, such as $\text{m}$, $\text{s}$, $\text{kg}$, $\text{N}$, or $\text{cm}$.
• Gradient: The change in a vertical variable divided by the change in a horizontal variable; it is mainly used when data are presented graphically.
• Outlier: A reading that is far away from the other readings and needs checking.

Worked Examples

Example 1: Calculate A Mean Time

A learner measures the time for a trolley to travel a fixed distance. The readings are $1.8\ \text{s}$, $1.9\ \text{s}$, and $1.8\ \text{s}$. Find the mean time.

Use the mean formula.

$$ \begin{aligned} \text{mean time} &= \frac{1.8\ \text{s} + 1.9\ \text{s} + 1.8\ \text{s}}{3} \\ &= \frac{5.5\ \text{s}}{3} \\ &= 1.83\ \text{s} \end{aligned} $$

The mean time is $1.83\ \text{s}$. If the stopwatch readings are being reported to one decimal place, this may be written as $1.8\ \text{s}$.

Example 2: Find The Range Of Length Readings

Three length readings are $24.6\ \text{cm}$, $24.8\ \text{cm}$, and $24.7\ \text{cm}$. Find the range.

$$ \begin{aligned} \text{range} &= \text{largest reading} - \text{smallest reading} \\ &= 24.8\ \text{cm} - 24.6\ \text{cm} \\ &= 0.2\ \text{cm} \end{aligned} $$

The range is $0.2\ \text{cm}$, so the readings are close together.

Example 3: Identify An Outlier

A learner records the following times for the same motion: $4.1\ \text{s}$, $4.2\ \text{s}$, $9.8\ \text{s}$, and $4.1\ \text{s}$. Identify the possible outlier and state what should be done.

The reading $9.8\ \text{s}$ is much larger than the others.

The learner should check the recording, repeat the trial if possible, and inspect the method before deciding whether to exclude it from the mean.

Example 4: Complete A Mean Column

The extension readings for a load are $3.0\ \text{cm}$, $3.1\ \text{cm}$, and $3.0\ \text{cm}$. Find the mean extension.

$$ \begin{aligned} \text{mean extension} &= \frac{3.0\ \text{cm} + 3.1\ \text{cm} + 3.0\ \text{cm}}{3} \\ &= \frac{9.1\ \text{cm}}{3} \\ &= 3.03\ \text{cm} \end{aligned} $$

The mean extension is about $3.0\ \text{cm}$ if recorded to one decimal place.

Common Mistakes

• Calling one reading a mean when no repeated readings were used.
• Finding the range by adding readings instead of subtracting smallest from largest.
• Mixing units in one mean, such as adding centimetres and metres without conversion.
• Deleting an outlier without checking or explaining the reason.
• Writing table headings without units.
• Reporting more decimal places than the instrument reading supports.
• Treating uncertainty as a mistake instead of a normal part of measurement.
• Drawing a conclusion that goes beyond the data.
• Confusing a trial with a variable.
• Averaging values from different experimental conditions.

Practice Tasks

1. Define data, reading, and trial in the context of a Physics experiment.
2. State two reasons why repeated readings are useful.
3. A learner records times of $5.2\ \text{s}$, $5.1\ \text{s}$, and $5.2\ \text{s}$. Calculate the mean time.
4. Find the range of the readings $18.4\ \text{cm}$, $18.6\ \text{cm}$, and $18.5\ \text{cm}$.
5. A thermometer has a smallest division of $1\ \text{C}$. State a simple estimate of the reading uncertainty when the scale is judged by eye.
6. Identify the outlier in the readings $6.0\ \text{N}$, $6.1\ \text{N}$, $12.8\ \text{N}$, and $6.0\ \text{N}$. Explain what should be done next.
7. Design a table for measuring the time for a trolley to travel different distances. Include three trials and a mean column.
8. Explain why the unit should appear in a table heading.
9. A learner calculates a mean from $10\ \text{cm}$, $12\ \text{cm}$, and $0.15\ \text{m}$. What must be done before calculating the mean?
10. Three groups measured the same length. Group A had range $0.1\ \text{cm}$, Group B had range $1.5\ \text{cm}$, and Group C had range $0.4\ \text{cm}$. Which group had the most consistent readings?
11. Write a conclusion from this data: when load increased from $1.0\ \text{N}$ to $4.0\ \text{N}$, mean extension increased from $2.0\ \text{cm}$ to $8.1\ \text{cm}$.
12. Explain why the 2022 examination format can guide practice tasks but should not change the official syllabus scope of this page.

Generated Question Layer

• Direct recall questions: Ask learners to define data, reading, trial, mean, range, uncertainty, variable, table, unit, and outlier.
• Table-completion questions: Provide repeated readings and ask learners to fill in mean and range columns.
• Error-detection questions: Ask learners to find missing units, mixed units, outliers, or unsuitable decimal places.
• Interpretation questions: Give a small data table and ask for a conclusion that quotes values and units.
• Practical-reasoning questions: Ask when to repeat a trial, when to investigate an outlier, and how uncertainty affects a conclusion.

Learner Aid Opportunities

• chart: Summary table comparing reading, trial, mean, range, uncertainty, and outlier.
• interactive: Mean and range calculator using learner-entered repeated readings.
• graph: Visual display showing close readings, spread readings, and an outlier on a number line.
• LLM tutor: Adaptive questioning that asks learners to inspect a table, calculate a mean, identify an outlier, and write a data-based conclusion.

Exam-Derived Signals

• No past-paper mappings have been reviewed for this specific Physics topic yet.
• The 2022 CSEE examination format may provide assessment-only signals for future practice design, especially data handling and interpretation tasks, but it does not define or expand the official syllabus scope used here.

Source And Review Notes

• Official syllabus status: extracted from the 2023 Physics syllabus as a Form I topic under data analysis, presentation, and report writing in Physics.
• Registry source: data/curricula/csee/physics/2023.json identifies the topic title, competence, form, source topic ID, and page path.
• Content authorship status: Explanations, examples, and practice tasks are original learner-facing prose written from the official syllabus topic and existing repo context.
• External enrichment status: no external web enrichment was used.
• Exam signal status: not mapped or reviewed in this milestone.
• Textbook status: no textbook wording was used.
• Review risk: A Physics reviewer should check the level of uncertainty language and decide whether local marking conventions prefer specific decimal-place rules.