The Central Idea

A statistical result is only as trustworthy as:

  1. the data that produced it;
  2. the methods used to summarize it;
  3. the way the result is displayed;
  4. the claims made from it.

A useful question is:

Does the display faithfully represent data collected by a process capable of supporting the conclusion?

Misleading Does Not Always Mean False

A graph can contain numerically correct values and still create a distorted visual impression.

Likewise, a statistic can be calculated correctly from data that are incomplete, biased, poorly measured, or irrelevant to the target population.

Statistical quality requires more than correct arithmetic.

Truncated Axes

A truncated axis begins at a value other than a natural baseline, often zero.

Suppose two values are

\[ 98 \]

and

\[ 100. \]

A bar chart whose vertical axis begins at \(97\) can make the second bar appear several times taller than the first even though the actual difference is only

\[ 100-98=2. \]

When a Truncated Axis Can Be Useful

Truncation is not automatically wrong.

It can be useful when:

Why Bar Charts Deserve Special Care

In a bar chart, the length of the bar often encodes magnitude directly.

If the baseline is shifted upward, the visible bar length can exaggerate relative differences.

For many bar charts, a zero baseline is therefore the clearest default.

Unequal Axis Intervals

Axis tick marks should represent consistent numerical intervals unless a special scale is explicitly identified.

A sequence such as

\[ 0,\ 10,\ 20,\ 100 \]

shown at equal visual spacing would distort numerical distance.

Compressed and Expanded Scales

A graph can make variation appear:

The scale should be chosen to clarify the data rather than force a desired visual narrative.

Changing the Aspect Ratio

The apparent steepness of a line can change when the graph is stretched vertically or horizontally.

The underlying numerical relationship is unchanged, but the visual impression can become much more dramatic or much flatter.

Three-Dimensional Effects

Three-dimensional bars, cylinders, cones, and perspective effects can make simple comparisons harder.

Perspective may cause:

Area Can Exaggerate Change

Suppose a pictorial symbol doubles in both width and height.

Its area becomes

\[ 2\cdot2=4 \]

times as large.

A quantity that doubled may therefore appear four times as large.

Volume Can Exaggerate Even More

If all three dimensions of a three-dimensional object double, volume changes by

\[ 2^3=8. \]

A value that doubled can appear eight times as large if volume is allowed to scale unintentionally.

Selective Time Windows

Choosing a particular start or end date can change the apparent trend.

For example, a market series may appear to be:

The time window should match the question being answered.

Cherry-Picking

Cherry-picking means selecting only observations, periods, subgroups, or comparisons that support a preferred conclusion.

Reliable analysis should use a defensible inclusion rule established by the study question or design.

Omitted Categories

A categorical display can be misleading if important categories are omitted.

If a chart shows only the three largest departments, the title should not imply that those are the only departments in the organization.

Pie Charts Must Represent a Whole

Pie-chart sectors should represent mutually compatible parts of one meaningful total.

If categories overlap, percentages may sum to more than

\[ 100\%. \]

A pie chart would then suggest a partition that does not actually exist.

Percentages Need Denominators

A percentage is meaningful only when its denominator is clear.

For example,

\[ 40\% \]

could mean:

The Denominator Can Change the Story

Suppose \(100\) people are contacted, \(80\) respond, and \(48\) say yes.

Among respondents:

\[ \frac{48}{80}=60\%. \]

Among everyone contacted:

\[ \frac{48}{100}=48\%. \]

Both calculations are correct, but they answer different questions.

Counts Versus Rates

Raw counts can mislead when groups differ in size.

Suppose Region A has \(200\) incidents among \(100{,}000\) people and Region B has \(150\) incidents among \(20{,}000\) people.

Region A has the larger count, but the rates are

\[ \frac{200}{100000}=0.002 \]

and

\[ \frac{150}{20000}=0.0075. \]

Region B has the larger rate.

Relative Change Versus Absolute Change

Suppose a quantity rises from \(2\) to \(4\).

The absolute increase is

\[ 4-2=2. \]

The relative increase is

\[ \frac{4-2}{2}=1=100\%. \]

Reports should distinguish absolute and percentage changes.

Percentage Points Versus Percent Change

Suppose a rate rises from \(20\%\) to \(25\%\).

The increase is

\[ 25\%-20\%=5 \]

percentage points.

The relative percent increase is

\[ \frac{25-20}{20}=0.25=25\%. \]

These are different quantities.

Aggregation Can Hide Important Patterns

Combining groups can conceal differences between subgroups.

An overall average may hide:

Simpson's Paradox

An important advanced phenomenon is Simpson's paradox.

A trend that appears within several separate groups can weaken, disappear, or reverse when the groups are combined.

This can occur because group sizes and lurking variables affect the aggregate result.

Why Group Structure Matters

Before reporting one overall mean, proportion, or correlation, ask whether relevant subgroups have materially different distributions.

Aggregation is not wrong, but it may answer a different question from subgroup analysis.

What Is Data Quality?

Data quality refers to whether data are suitable for the intended analytical purpose.

Useful dimensions include:

Accuracy

Accuracy concerns whether recorded values correspond reasonably to the quantities they are intended to measure.

Sources of inaccuracy include:

Completeness

Completeness concerns whether necessary observations and variables are present.

Missing data can reduce effective sample size and can introduce bias if the missingness is systematic.

Consistency

Consistency means that values follow compatible definitions and coding rules.

Problems include:

Validity

Validity asks whether a value satisfies the logical rules of the variable.

Examples:

Timeliness

Data can be accurate but too old for the decision being made.

Timeliness asks whether the data reflect the relevant period.

This is especially important in rapidly changing systems.

Representativeness

A dataset is representative when the observed cases adequately reflect the population relevant to the intended conclusion.

Large datasets can still be unrepresentative.

Traceability

Traceability means being able to determine:

Traceability supports reproducibility and auditability.

Missing Data

Missing values require explicit treatment.

Common missing-data codes include:

Special codes must not accidentally be treated as real measurements.

Missing Completely at Random

In introductory terminology, data are missing completely at random when the probability of missingness is unrelated to observed or unobserved values.

This is a strong condition and should not be assumed casually.

Systematic Missingness

Missingness can be related to the phenomenon being studied.

For example, people with very high incomes may be less willing to report income.

In such cases, simply analyzing complete responses can bias the result.

Nonresponse Bias

Nonresponse bias occurs when people who do not respond differ systematically from those who do.

A high response count alone does not eliminate this problem.

Undercoverage

Undercoverage occurs when some members of the target population have little or no chance of being included.

An online-only survey may underrepresent people without reliable internet access.

Voluntary-Response Bias

A voluntary-response sample allows people to choose themselves into the study.

People with unusually strong opinions may be more likely to participate, making the sample unrepresentative.

Convenience Samples

A convenience sample uses cases that are easy to reach.

Convenience can reduce cost, but it can also create serious selection bias when accessibility is related to the variables being studied.

Measurement Error

Measurement error is the difference between the recorded value and the intended underlying quantity.

It can arise from:

Systematic Versus Random Measurement Error

Random measurement error introduces unpredictable variation.

Systematic error pushes measurements consistently in one direction.

For example, a scale that always reads \(2\) kg too high introduces systematic error.

Question Wording

Survey responses can depend on how a question is phrased.

Leading, loaded, ambiguous, or double-barreled questions can alter the distribution of responses.

Order Effects

Earlier questions can influence answers to later questions.

This is called an order effect.

Survey design should therefore consider question sequence as part of the measurement process.

Duplicate Records

Duplicate cases can inflate frequencies, counts, and apparent sample size.

Before analysis, determine whether repeated records represent:

Unit Errors

Mixing units can create extreme values that look like outliers.

For example, a dataset of heights in centimeters may contain one value entered in meters:

\[ 1.75 \]

instead of

\[ 175. \]

Data validation should catch such inconsistencies before interpretation.

Impossible and Implausible Values

Analysts should distinguish:

Impossible values can often be corrected or excluded with clear justification. Implausible values require investigation.

Outliers Are Data-Quality Signals, Not Automatic Errors

An outlier can reveal:

The appropriate response is investigation.

Data Cleaning Should Be Documented

Any cleaning decision should be reproducible.

Document:

Do Not Clean Toward a Desired Result

Data cleaning should improve validity and consistency.

It should not be used to selectively remove observations that weaken a preferred conclusion.

Metadata Matters

Good datasets should be accompanied by information describing:

These supporting descriptions are often called metadata.

Reproducibility

A statistical analysis is more trustworthy when another analyst can reproduce the result from the same data and documented procedures.

Reproducibility requires:

Visual Integrity

A useful display should:

Statistical Integrity

A useful analysis should:

Example: Misleading Bar Chart

Suppose two departments have satisfaction rates of

\[ 91\% \]

and

\[ 94\%. \]

If the vertical axis begins at \(90\%\), the second bar may appear several times taller.

The actual difference is

\[ 94\%-91\%=3 \]

percentage points.

Example: Misleading Denominator

A company reports that "80\% of customers recommend the product."

If this means \(80\) of \(100\) respondents but only \(100\) of \(10{,}000\) customers responded, the statement requires context.

The recommendation rate among respondents is \(80\%\), but nonresponse may limit generalization to all customers.

Example: Data Quality Failure

Suppose a dataset records delivery times in minutes, but one source system supplies seconds.

A value of

\[ 900 \]

seconds may be interpreted incorrectly as

\[ 900 \]

minutes unless units are standardized.

The correct conversion is

\[ \frac{900}{60}=15\text{ minutes}. \]

Example: Selection Bias

Suppose a university measures campus satisfaction by surveying only students attending a voluntary alumni-networking event.

The sample may differ systematically from the full student population.

A large number of responses does not eliminate this selection problem.

Common Error: Trusting a Polished Graphic Automatically

Professional design does not guarantee statistical quality.

Always inspect:

Common Error: Assuming Large Data Means Good Data

A dataset with millions of observations can still be systematically biased.

More data reduce some forms of random variation, but they do not automatically correct:

Common Error: Treating Missing Values as Zero

Missing and zero are different concepts.

A zero may be a legitimate observed quantity.

A missing value means the quantity was not observed or not applicable.

Common Error: Ignoring Changing Definitions

A time series may become incomparable if the definition of a variable changes.

If "active customer" is defined differently before and after a policy change, apparent trend changes may partly reflect the definition rather than behavior.

Common Error: Reporting Precision the Data Do Not Support

If measurements are recorded only to the nearest whole unit, reporting a mean to six decimal places can create false precision.

Numerical reporting should reflect measurement and sampling precision.

Common Error: Hiding Uncertainty

A sample statistic is not generally identical to the population parameter.

Later statistical inference will quantify uncertainty more formally, but even descriptive reports should avoid presenting sample estimates as exact population facts.

Common Error: Causal Language from Descriptive Data

A graph or correlation may show association.

It does not automatically justify statements such as:

"X causes Y."

Study design determines whether causal interpretation is defensible.

A Reliable Display Review Procedure

  1. Read the title carefully.
  2. Identify the variables and units.
  3. Inspect both axes and baselines.
  4. Check whether spacing reflects actual numerical intervals.
  5. Determine whether values are counts, rates, percentages, or indices.
  6. Identify the denominator.
  7. Check the time period and starting point.
  8. Look for omitted categories or selective ranges.
  9. Ask whether visual area or volume exaggerates magnitude.
  10. Compare the visual impression with the actual numerical differences.

A Reliable Data-Quality Review Procedure

  1. Define the target population and sample.
  2. Review the sampling frame and selection process.
  3. Check variable definitions and units.
  4. Inspect missing-value patterns.
  5. Check for duplicates and invalid values.
  6. Investigate unusual observations.
  7. Verify time periods and category definitions.
  8. Review measurement and survey procedures.
  9. Document cleaning and transformation rules.
  10. Confirm that the data support the intended claim.

Practice 1: Truncated Axis

A bar chart compares values \(48\) and \(50\), but the vertical axis begins at \(47\).

Why can the visual difference be misleading?

Practice 2: Denominator

A survey contacts \(200\) people. Only \(100\) respond, and \(70\) of those respondents approve.

Find the approval percentage among respondents and among everyone contacted.

Practice 3: Percentage Points

A rate rises from \(30\%\) to \(36\%\).

Find the increase in percentage points.

Practice 4: Relative Percent Increase

Using the same change from \(30\%\) to \(36\%\), find the relative percent increase.

Practice 5: Count or Rate?

Hospital A records \(120\) infections among \(12{,}000\) patients. Hospital B records \(80\) infections among \(4{,}000\) patients.

Which hospital has the higher infection rate?

Practice 6: Missing Data

A dataset uses \(-999\) to represent missing income values.

Why would including \(-999\) in the mean be incorrect?

Practice 7: Unit Consistency

Most heights are recorded in centimeters, but one value is \(1.82\) because it was entered in meters.

What data-quality problem is present?

Practice 8: Sample Bias

A website asks visitors to voluntarily vote on whether they like the site.

What sampling problem may occur?

Practice 9: Outlier Handling

One observation is far larger than the rest.

What should an analyst do before deleting it?

Practice 10: Full Review

A chart reports a \(200\%\) increase but does not show the original value, denominator, sample size, or time period.

Name at least three pieces of information needed before interpreting the claim confidently.

Practice Answers

Practice 1

Beginning the axis at \(47\) removes most of the baseline and magnifies a numerical difference of only

\[ 50-48=\boxed{2}. \]

Practice 2

Among respondents:

\[ \frac{70}{100}=0.70=\boxed{70\%}. \]

Among everyone contacted:

\[ \frac{70}{200}=0.35=\boxed{35\%}. \]

Practice 3

\[ 36\%-30\% = \boxed{6\text{ percentage points}}. \]

Practice 4

\[ \frac{36-30}{30} = \frac{6}{30} = 0.20 = \boxed{20\%}. \]

Practice 5

Hospital A:

\[ \frac{120}{12000}=0.01=1\%. \]

Hospital B:

\[ \frac{80}{4000}=0.02=2\%. \]

Hospital B has the higher infection rate.

Practice 6

The value \(-999\) is a code for missingness rather than a measured income. Treating it as a real value would distort the mean.

Practice 7

The dataset contains a unit inconsistency. The value should be converted to the common unit before analysis.

Practice 8

The study may suffer from voluntary-response bias because people with stronger opinions may be more likely to participate.

Practice 9

Investigate the observation's source, units, data-entry accuracy, measurement conditions, and whether it represents a valid rare case or a different subgroup.

Practice 10

Useful missing information includes:

What to Remember

Where We Go Next

This lesson completes the core descriptive-statistics sequence by combining statistical presentation with data-quality judgment.

The next stage of statistical study builds on these foundations to examine formal sampling behavior, estimation, uncertainty, and inference.