Data Investigations

A statistical investigation is a systematic process used to answer a question by collecting, organizing, analysing, and interpreting data.

A strong investigation does more than produce graphs or calculate averages. It begins with a meaningful question and uses evidence to reach a justified conclusion.

A typical statistical investigation follows a cycle:

Question → Plan → Collect → Organize → Analyze → Interpret → Conclude → Communicate

The process is often described as a cycle because conclusions can lead to new questions and further investigations.

For example, a school might investigate:

"How many hours of sleep do Grade 10 students typically get on a school night?"

To answer this well, investigators must decide who to survey, what information to collect, how to summarize the data, and what conclusions the evidence supports.

Starting with a Statistical Question

Every investigation begins with a research question.

A useful statistical question:

• can be answered using data
• expects variation in the answers
• identifies what will be measured
• identifies the population or group of interest
• is specific enough to investigate

Consider:

"How tall is Daniel?"

This is not really a statistical question because it asks for one measurement.

Compare it with:

"What is the typical height of Grade 9 students at our school?"

This is statistical because student heights vary and many measurements are needed.

Statistical Questions Expect Variation

Variation is central to statistics.

Consider:

"How many siblings does Maria have?"

There is one answer.

But:

"How many siblings do students in our class have?"

will probably produce several different answers.

The variation in those answers can then be analysed statistically.

Improving a Research Question

A question such as:

"Do students sleep enough?"

is too vague for a strong investigation.

We need to define:

• which students
• what "enough" means
• what period is being studied
• how sleep will be measured

A stronger question might be:

"How many hours of sleep do Grade 9 students at our school report getting on a typical school night?"

Now the variable and population are much clearer.

Population and Sample

The population is the entire group we want to understand.

The sample is the smaller group from which we actually collect data.

For example:

Research question:

"How much time do students at our school spend travelling to school?"

Population:

all students at the school

Sample:

120 students selected for the investigation

Usually it is impractical to collect data from every member of a large population, so we use a sample.

Choosing a Good Sample

A useful sample should be reasonably representative of the population.

Suppose a school has students from Grades 7–12.

If we survey only Grade 12 students about travel time, the results may not represent the whole school.

A better sample might include students from every grade.

A representative sample makes it more reasonable to generalize findings from:

sample → population

Sample Size

Larger samples generally provide more information and tend to reduce the effects of random sampling variation.

Suppose a school has 2,000 students.

Surveying:

5 students

would provide very limited evidence.

Surveying:

300 appropriately selected students

would generally provide much stronger evidence.

However:

a large biased sample can still produce poor results.

Sample quality matters as well as sample size.

Sampling Methods

Several methods can be used to select participants.

Simple Random Sample

Members of the population are selected randomly so that each has an equal chance of selection.

For example, assign every student a number and use a random-number generator to select participants.

Systematic Sample

Choose members at regular intervals after a suitable starting point.

For example:

survey every 10th student on an appropriately ordered list after selecting a random starting position.

Stratified Sample

Divide the population into important groups and sample from each.

For example, if a school has six grade levels, students could be sampled from every grade, ideally in proportions reflecting the school's population.

Convenience Sample

Collect data from whoever is easiest to reach.

For example:

surveying students sitting near you at lunch.

This is easy, but it may introduce substantial bias.

Bias in Sampling

Bias is a systematic tendency for the data or conclusions to favour certain outcomes.

Suppose we investigate:

"How much exercise do students at our school get?"

but collect data only from students leaving the gym.

The sample is likely to overrepresent physically active students.

This is selection bias.

Planning the Investigation

Before collecting data, prepare a plan.

A good plan identifies:

• the research question
• the population
• the sample
• the sampling method
• the variables
• how each variable will be measured
• how the data will be recorded
• possible sources of bias
• ethical or privacy considerations
• statistical methods likely to be used

Planning first helps prevent problems that may be impossible to fix after the data have been collected.

Variables

A variable is a characteristic that can take different values.

Examples include:

• height
• age
• travel time
• favourite subject
• number of siblings
• test score
• temperature

Different types of variables require different methods of analysis.

Categorical Data

Categorical data place observations into groups or categories.

Examples include:

• eye colour
• transportation method
• favourite school subject
• type of device used

Possible transportation data might be:

bus, car, bicycle, walk, train

These are categories rather than numerical measurements.

Numerical Data

Numerical data consist of numbers representing counts or measurements.

Examples include:

• height
• mass
• temperature
• test score
• number of siblings
• travel time

Numerical data can be further classified as discrete or continuous.

Discrete Data

Discrete data usually come from counting and take separate values.

Examples:

• number of siblings
• number of goals scored
• number of books read
• number of students absent

A student cannot normally have:

2.7 siblings

so the values occur in separate steps.

Continuous Data

Continuous data come from measurements and can theoretically take any value within an interval.

Examples include:

• height
• mass
• time
• distance
• temperature

A person's height might be:

171 cm

171.4 cm

171.43 cm

depending on the precision of the measuring instrument.

Primary and Secondary Data

Primary data are collected directly for the investigation.

Examples:

• conducting a survey
• measuring student heights
• recording temperatures
• performing an experiment

Secondary data were originally collected by someone else.

Examples:

• government statistics
• published research
• databases
• school records
• historical weather data

Both can be useful, but the source and quality of secondary data should be evaluated.

Designing a Survey

Survey questions should be:

• clear
• neutral
• specific
• relevant
• easy to interpret consistently

Poor question:

"Don't you agree that students receive far too much homework?"

This is leading.

Better:

"Approximately how much time do you spend on homework on a typical school night?"

The second question collects information without strongly encouraging a particular response.

Choosing Response Options

Response options must also be designed carefully.

Suppose the question asks:

How long do you spend travelling to school?

Possible categories might be:

• less than 15 minutes
• 15–29 minutes
• 30–44 minutes
• 45–59 minutes
• 60 minutes or more

The categories should not overlap.

A student taking exactly 30 minutes should clearly know which category to choose.

Collecting Measurements

When collecting measurements, investigators should use consistent procedures.

Suppose students' heights are being compared.

Everyone should ideally be measured:

• with the same type of instrument
• using the same units
• using the same procedure
• to an appropriate level of precision

Otherwise, differences in measurement method may introduce unwanted variation or bias.

Organizing Raw Data

Raw data are the original observations before they have been summarized.

Suppose 20 students report the number of books they read last month:

2, 4, 1, 3, 2, 5, 0, 3, 2, 4, 1, 2, 6, 3, 2, 1, 4, 2, 3, 2

This list contains useful information, but patterns are difficult to see.

A frequency table makes the data easier to understand.

Organizing data is an important step between collection and analysis.

Frequency and Relative Frequency

Frequency tells us how often a value occurs.

Relative frequency tells us the proportion of observations belonging to a value or category.

For example, 7 of the 20 students read two books.

Relative frequency:

7/20 = 0.35

or:

35%

Relative frequencies are especially useful when comparing groups of different sizes.

Choosing an Appropriate Graph

Different data require different graphical displays.

The graph should be chosen because it communicates the relevant feature of the data—not simply because it looks attractive.

Bar Graphs and Histograms

A bar graph is commonly used for categorical data.

The bars are separated because the categories are distinct.

A histogram is used for numerical data grouped into intervals.

The bars normally touch because the intervals represent adjacent ranges of numerical values.

These graphs may look similar, but they represent different types of information.

Measures of Centre

Measures of centre describe a typical or central value.

The main measures are:

mean

median

mode

The mean is:

mean = sum of values / number of values

The median is the middle value after ordering the data.

The mode is the most frequently occurring value.

Calculating the Mean

Consider:

4, 6, 7, 8, 10

Mean:

(4 + 6 + 7 + 8 + 10) / 5

= 35/5

= 7

Calculating the Median

For:

4, 6, 7, 8, 10

the middle value is:

7

Therefore:

median = 7

If there are an even number of observations, take the mean of the two middle values.

For:

4, 6, 8, 10

median = (6 + 8)/2

= 7

Calculating the Mode

Consider:

2, 3, 3, 3, 5, 6, 8

The value occurring most frequently is:

3

Therefore:

mode = 3

A data set may have:

• one mode
• more than one mode
• no mode

Which Measure of Centre Should We Use?

The mean is useful because it uses every value.

However, it can be strongly affected by extreme values.

Consider monthly incomes:

$2,500
$2,700
$2,800
$3,000
$20,000

Mean:

(2500 + 2700 + 2800 + 3000 + 20000)/5

= $6,200

Median:

$2,800

The median better represents the typical income in this data set because the $20,000 value strongly affects the mean.

Measures of Spread

Centre alone does not fully describe data.

Consider:

Data Set A:

48, 49, 50, 51, 52

Data Set B:

10, 30, 50, 70, 90

Both have a mean of:

50

But the distributions are very different.

Data Set B is much more spread out.

Measures of spread describe this variability.

Range

The simplest measure of spread is the range.

range = maximum − minimum

For:

10, 30, 50, 70, 90

range = 90 − 10

= 80

For:

48, 49, 50, 51, 52

range = 52 − 48

= 4

The second data set is much less variable.

Quartiles and Interquartile Range

Quartiles divide ordered data into sections.

Q₁ = lower quartile

Q₂ = median

Q₃ = upper quartile

The interquartile range is:

IQR = Q₃ − Q₁

The IQR describes the spread of the middle 50% of the data.

It is less strongly affected by extreme values than the range.

Box Plots

A box plot summarizes a distribution using values such as:

• minimum
• Q₁
• median
• Q₃
• maximum

Depending on the convention used, unusually distant observations may be plotted separately as outliers rather than used as the whisker endpoints.

Box plots are particularly useful for comparing distributions between groups.

Outliers

An outlier is an observation unusually distant from the rest of the data.

Suppose travel times are:

12, 14, 15, 16, 17, 18, 19, 85 minutes.

The value:

85

is much larger than the others.

Before removing an outlier, ask why it occurred.

It could be:

• a recording error
• a measurement error
• an unusual but genuine observation
• evidence of an important subgroup

Outliers should be investigated rather than automatically deleted.

Analysing Relationships

Sometimes an investigation examines two numerical variables.

For example:

Is there a relationship between hours studied and test score?

A scatter plot is useful.

The pattern may show:

• positive correlation
• negative correlation
• weak correlation
• little or no correlation
• unusual points or outliers

Correlation Does Not Establish Causation

Suppose students who sleep longer tend to have higher test scores.

This shows an association.

It does not necessarily prove:

more sleep caused the higher scores.

Other variables might include:

• study habits
• stress
• health
• age
• workload
• attendance

Statistical conclusions should not claim more than the investigation can establish.

Comparing Groups

A statistical investigation may compare two or more groups.

For example:

Do Grade 9 and Grade 11 students have different travel times to school?

Useful comparisons could include:

• mean
• median
• range
• IQR
• distribution shape
• unusual observations

A strong analysis does not simply state:

"Group A has a higher median."

It might say:

"Grade 11 students had a median travel time of 32 minutes compared with 24 minutes for Grade 9 students, suggesting longer typical travel times in the Grade 11 sample."

The numerical evidence strengthens the conclusion.

Distribution Shape

The shape of a distribution can reveal important information.

A distribution might be:

• approximately symmetric
• skewed to the right
• skewed to the left
• clustered
• bimodal
• affected by outliers

For a strongly skewed distribution, the median may describe the centre more appropriately than the mean.

Evidence-Based Conclusions

A statistical conclusion should directly answer the original research question.

Suppose the question was:

"How many hours of sleep do Grade 9 students at our school typically get?"

Imagine the sample produced:

mean = 7.1 hours
median = 7.3 hours
IQR = 1.2 hours

A reasonable conclusion might be:

The sampled Grade 9 students typically reported about seven hours of sleep per school night. The median was 7.3 hours and the middle 50% of responses spanned 1.2 hours, suggesting moderate variation in reported sleep time.

This conclusion uses actual evidence rather than vague language.

Avoiding Conclusions That Are Too Strong

Suppose we surveyed 50 Grade 9 students.

It would usually be too strong to conclude:

"All Grade 9 students sleep 7.3 hours."

A more appropriate statement is:

"The sample suggests that a typical Grade 9 student in the population represented by this sample sleeps about seven hours."

Statistical conclusions usually involve uncertainty.

Sample-to-Population Conclusions

When using a sample to make conclusions about a population, consider:

• sample size
• sampling method
• representativeness
• measurement quality
• missing responses
• possible bias

The weaker these features are, the more cautious the conclusion should be.

Experimental and Observational Investigations

An observational study measures variables without deliberately assigning treatments.

Example:

Measure students' usual sleep duration and their test scores.

An experiment deliberately changes a variable and observes the effect.

Example:

Randomly assign participants to two different study methods and compare their performance.

Well-designed randomized experiments generally provide stronger evidence about causation than observational studies.

Reliability

Reliability refers to the consistency of measurements or results.

Suppose a scale gives:

52.1 kg
57.4 kg
49.8 kg

for the same object measured seconds apart.

The measurement process is not very reliable.

Reliable methods produce reasonably consistent measurements under the same conditions.

Validity

Validity concerns whether an investigation actually measures what it claims to measure.

Suppose we want to investigate:

student fitness

but measure only:

height

Height can be measured reliably, but it is not a valid measure of overall fitness.

A measurement can therefore be:

reliable but not valid.

Good investigations aim for both.

Accuracy and Precision

These concepts are also useful when collecting measurements.

Accuracy describes how close a measurement is to the true or accepted value.

Precision describes how closely repeated measurements agree with one another.

A set of measurements can be:

• accurate and precise
• accurate on average but not very precise
• precise but inaccurate
• neither accurate nor precise

Sources of Error and Bias

Potential problems should be considered throughout an investigation.

These can include:

• selection bias
• voluntary response bias
• non-response bias
• leading questions
• inaccurate measurements
• inconsistent procedures
• recording mistakes
• missing data
• small sample size
• poorly defined variables

Identifying limitations does not make an investigation worthless.

It helps us understand how strongly we can interpret its results.

Ethics and Privacy

Statistical investigations involving people should consider ethical issues.

Depending on the investigation, this may involve:

• informed participation
• avoiding unnecessary personal information
• keeping responses confidential
• reporting results anonymously
• storing data securely
• allowing participants to decline participation

Questions about highly personal information should only be collected when genuinely necessary.

Presenting Results

The final stage of an investigation is communicating the findings.

A clear report might include:

Research question

What was investigated?

Method

How were the data collected?

Sample

Who or what was studied?

Results

What statistics and graphs summarize the data?

Interpretation

What patterns were found?

Conclusion

What does the evidence suggest?

Limitations

What reduces confidence in the conclusion?

Tables and Graphs

Tables are useful when exact values matter.

Graphs are useful when patterns and comparisons matter.

Often, using both provides the clearest presentation.

However, every table or graph should have:

• a meaningful title
• clearly labelled variables
• appropriate units
• sensible scales
• readable categories
• accurate values

Avoid decorative features that make the data harder to understand.

Choosing Evidence for a Presentation

Not every calculation needs to appear in the final presentation.

Choose statistics that directly help answer the research question.

For example, if comparing travel times between two groups, useful information might include:

• sample sizes
• medians
• IQRs
• box plots

Listing dozens of unrelated calculations can make the main result harder to understand.

Effective communication is selective without being misleading.

Example Investigation: Student Travel Time

Suppose the research question is:

"How long does it typically take students at our school to travel to school?"

Plan

Population:

All students at the school.

Sample:

150 students selected across different grade levels.

Variable:

Travel time in minutes.

Method:

Anonymous survey asking students for their typical one-way travel time.

Collect

Students report their travel times.

Organize

The data are entered into a table and checked for obvious recording errors.

Analyze

Calculate:

• mean
• median
• range
• quartiles
• IQR

Create:

• histogram
• box plot

Interpret

Suppose:

median = 28 minutes

mean = 34 minutes

Several students report journeys longer than 90 minutes.

The mean being higher than the median suggests that long travel times may be pulling the mean upward.

Conclude

A suitable conclusion might be:

Students in the sample typically travelled about 28 minutes to school. The distribution was right-skewed because a smaller number of students reported much longer journeys, which raised the mean to 34 minutes.

This conclusion combines numerical evidence with interpretation.

Example Investigation: Study Time and Test Scores

Research question:

"Is there an association between weekly study time and mathematics test scores?"

Collect two variables from each participant:

• weekly study time
• mathematics score

Use a scatter plot.

If the points generally rise from left to right, the data show a positive association.

However, the conclusion should be:

"Students who reported more study time tended to have higher scores."

It should not automatically be:

"Studying longer causes higher scores."

The investigation may be observational and other variables may be involved.

Improving an Investigation

After completing an investigation, ask how it could be improved.

Possible improvements include:

• increasing sample size
• using random sampling
• including underrepresented groups
• rewriting unclear questions
• using more accurate instruments
• repeating measurements
• collecting data over a longer period
• recording additional relevant variables
• reducing missing responses

Evaluation is part of good statistical practice.

Repeating the Investigation

Suppose another class repeats the same investigation and obtains slightly different results.

That is not necessarily a problem.

Different random samples naturally produce different statistics.

For example:

Sample A median = 27 minutes

Sample B median = 29 minutes

Sample C median = 28 minutes

Sampling variation is expected.

Statistics help us make conclusions while recognizing that uncertainty.

The Statistical Investigation Cycle

The complete process can be summarized as:

Ask

Develop a clear statistical question.

Plan

Identify the population, sample, variables, and collection method.

Collect

Gather relevant, reliable data.

Organize

Clean, classify, and summarize the observations.

Analyze

Use appropriate graphs and statistics.

Interpret

Identify meaningful patterns and relationships.

Conclude

Answer the original question using evidence.

Communicate

Present the results clearly and acknowledge limitations.

Then new questions may emerge, beginning another investigation.

Common Mistakes

Collecting data before developing a clear question

Without a focused question, it may be unclear which data are actually needed.

Using a convenience sample without considering bias

Easy-to-reach participants may not represent the population.

Assuming a large sample cannot be biased

A large biased sample is still biased.

Using the wrong type of graph

For example, a histogram and a bar graph serve different purposes.

Reporting only the mean

Centre alone does not describe the distribution. Spread and shape may also matter.

Automatically removing outliers

An unusual observation should first be investigated.

Confusing correlation with causation

An observational relationship does not automatically establish cause and effect.

Ignoring units

A mean of "24" is incomplete if the quantity should be reported as 24 minutes.

Making conclusions stronger than the evidence

A small or biased sample requires cautious conclusions.

Ignoring limitations

Every real investigation has limitations. Identifying them improves the quality of the analysis.

Creating graphs before checking the data

Errors, missing values, and incorrect units should be investigated before analysis.

Key Terms

Statistical investigation — a systematic process of using data to answer a question.

Statistical question — a question that anticipates variability and can be investigated using data.

Population — the complete group about which information is sought.

Sample — a subset of the population from which data are collected.

Representative sample — a sample that reasonably reflects important characteristics of the population.

Variable — a characteristic that can take different values.

Categorical data — data consisting of groups or categories.

Numerical data — data represented by numerical values.

Discrete data — numerical data based on separate countable values.

Continuous data — numerical data that can take values throughout an interval.

Primary data — data collected directly for the investigation.

Secondary data — data previously collected by another person or organization.

Frequency — the number of times a value or category occurs.

Relative frequency — the proportion of observations belonging to a value or category.

Mean — the sum of the values divided by the number of observations.

Median — the middle value of ordered data.

Mode — the most frequently occurring value.

Range — maximum value minus minimum value.

Interquartile range (IQR) — Q₃ − Q₁, describing the spread of the middle 50% of the data.

Outlier — an observation unusually distant from the overall distribution.

Correlation — a statistical association between variables.

Bias — a systematic influence that can distort data or conclusions.

Reliability — the consistency of a measurement or procedure.

Validity — the extent to which a method measures what it is intended to measure.

Sampling variation — natural differences between statistics obtained from different samples.

Key Takeaways

• A statistical investigation begins with a clear question that expects variability in the data.
• The population is the complete group of interest, while the sample is the group actually studied.
• A representative sample is essential when making conclusions about a larger population.
• Sample size matters, but a large biased sample can still produce misleading results.
• Data may be categorical or numerical, and numerical data may be discrete or continuous.
• Data collection methods should be consistent, relevant, and as unbiased as possible.
• Raw data should be organized before analysis.
• Graphs should be selected according to the type of data and the purpose of the investigation.
• Mean, median, mode, range, and IQR describe different features of a distribution.
• Outliers should be investigated rather than automatically removed.
• Scatter plots can reveal associations between numerical variables, but correlation does not by itself establish causation.
• Conclusions should directly answer the research question and cite numerical or graphical evidence.
• The strength of a conclusion should reflect the quality of the sample and data.
• Limitations, bias, uncertainty, reliability, and validity should be acknowledged.
• Statistical investigations form a cycle in which conclusions often lead to new questions and improved investigations.

Check Your Understanding

1. Explain what makes a question a statistical question. Give one statistical and one non-statistical example.
2. A school wants to investigate how long students spend on homework each night. Identify the population, a possible sample, and the main variable.
3. Explain why surveying only students in the library might produce a biased estimate of homework time for the whole school.
4. Distinguish between categorical, discrete numerical, and continuous numerical data. Give an example of each.
5. Explain the difference between primary data and secondary data.
6. A survey asks, "Don't you agree that students receive too much homework?" Explain the problem and rewrite the question more neutrally.

7. Calculate the mean, median, mode, and range of:

   4, 5, 5, 6, 7, 7, 7, 8, 11

8. Two classes both have a mean test score of 70%. Class A has scores tightly grouped around 70%, while Class B has scores ranging from 30% to 100%. Explain why reporting only the means would hide an important difference.
9. Explain why a histogram is more appropriate than a bar graph for displaying grouped continuous measurements such as student heights.
10. A data set contains:

12, 13, 14, 15, 15, 16, 17, 74

Identify the likely outlier and explain what should be done before deciding whether to remove it.

11. A scatter plot shows a positive relationship between weekly study time and test scores. State an appropriate conclusion and explain why the graph alone cannot establish causation.
12. A sample of 10 students has a median travel time of 25 minutes. Another carefully selected sample of 300 students has a median of 31 minutes. Which result would generally provide stronger evidence about the school population, and what else would you need to know before deciding?
13. Design a statistical investigation to answer:

"How much sleep do students at our school get on a typical school night?"

Identify the population, sample, sampling method, variable, data-collection method, and at least two possible sources of bias.

14. An investigation finds that sampled students have a median travel time of 28 minutes, a mean of 35 minutes, and several travel times greater than 90 minutes. Explain what these results suggest about the shape of the distribution.
15. Imagine that you have completed a statistical investigation. Describe what should be included in your final report so that another person can understand what you investigated, how you collected the data, what you found, how strong the evidence is, and what limitations remain.