Chapter 3

Question 1

Definition of an Outlier

Answer 1

An outlier is a value that lies significantly outside the pattern of the data.

Standard formula to detect outliers:
Outlier < Q₁ - 1.5(Q₃ - Q₁)
or
Outlier > Q₃ + 1.5(Q₃ - Q₁)

Alternative method:
Outlier < 𝑥̄ - 2σ
or
Outlier > 𝑥̄ + 2σ

Question 2

Removing Anomalies (Cleaning Data)

Answer 2

The process of removing errors or irrelevant data is called cleaning the data.
Steps in cleaning data:
Identify outliers using IQR or standard deviation.
Decide whether to keep or remove:
Keep if valid.
Remove if an error.
Justify your decision.

Question 3

Box Plots

Answer 3

A box plot represents:

Minimum (non-outlier)
Lower Quartile (Q₁)
Median (Q₂)
Upper Quartile (Q₃)
Maximum (non-outlier)
Outliers plotted separately

Question 4

Comparing Box Plots

Answer 4

Compare the medians (center of data).
Compare the IQRs (spread of middle 50%).
Check for skewness.
Look for outliers.

Question 5

Cumulative Frequency & Quartiles

Answer 5

A cumulative frequency graph helps estimate:

Median at n/2
Lower Quartile (Q₁) at n/4
Upper Quartile (Q₃) at 3n/4

Question 6

Interquartile Range (IQR)

Answer 6

IQR = Q₃ - Q₁

Measures spread of middle 50% of data.
Not affected by outliers.

Question 7

Percentile Range

Answer 7

10th to 90th percentile range:
P₉₀ - P₁₀

Better for comparing spread than just using range.

Question 8

Histograms

Answer 8

A histogram represents continuous data.

Formula for Frequency Density:
Frequency Density = Frequency ÷ Class Width

Question 9

Properties of Histograms

Answer 9

No gaps between bars.
Area of bar is proportional to frequency.
Used for continuous data.

Question 10

Estimating from a Histogram

Answer 10

Find total area under the histogram.
Estimate frequency of a subset using part of an area.

Question 11

Comparing Histograms

Answer 11

When comparing histograms:

Use frequency density, not just bar height.
Compare spread.
Compare peak values.

Question 12

Skewness

Answer 12

Symmetric (mean ≈ median ≈ mode)
Positive skew (mean > median > mode)
Negative skew (mode > median > mean)
Formula for Skewness:
Skewness = 3 × (𝑥̄ - Median) ÷ σ

Question 13

Comparing Distributions

Answer 13

Compare location (median, mean).
Compare spread (IQR, range).
Look for skewness.
Comment on outliers.

Question 14

Exam Tips for Data Representation

Answer 14

Use correct formulas.
Check for outliers.
Use interpolation for quartiles.
For histograms, use frequency density.
Use box plots for comparisons.
Cumulative frequency graphs estimate medians and quartiles.