Data Presentation and Interpretation
Data Presentation and Interpretation
This topic covers the statistical tools for summarising, displaying and interpreting data: measures of location, measures of spread, data coding, and identifying outliers. These skills underpin all statistical analysis.
Measures of Central Tendency (Location)
| Measure | Definition | Formula / Method |
|---|---|---|
| Mean (x̄) | Sum of values divided by count | x̄ = Σx / n |
| Median | Middle value when data is ordered | (n+1)/2 th value |
| Mode | Most frequently occurring value | Observation; may not be unique |
For grouped data (frequency tables), the mean is estimated using midpoints:
x̄ = Σfx / Σf where x is the class midpoint and f is the frequency.
The median class is found by locating n/2 in the cumulative frequency. Linear interpolation gives an estimate within the class.
Worked Example: Data: 3, 5, 7, 7, 8, 12.
Mean = (3+5+7+7+8+12)/6 = 42/6 = 7
Median = average of 3rd and 4th values = (7+7)/2 = 7
Mode = 7 (appears twice)
Measures of Spread (Dispersion)
| Measure | Formula |
|---|---|
| Range | Largest - Smallest |
| Interquartile range (IQR) | Q₃ - Q₁ |
| Variance (σ² or s²) | Σ(x - x̄)² / n or Σx²/n - x̄² |
| Standard deviation (σ or s) | √(variance) |
The variance formula Σx²/n - (Σx/n)² is often easier to compute. In this form:
Variance = mean of squares - square of mean
Worked Example: Data: 2, 4, 6, 8, 10. Mean = 6.
Σx² = 4 + 16 + 36 + 64 + 100 = 220
Variance = 220/5 - 6² = 44 - 36 = 8
Standard deviation = √8 ≈ 2.83
Quartiles and Percentiles
The lower quartile Q₁ is the value below which 25% of the data falls; the upper quartile Q₃ is the value below which 75% falls.
For n data points (ordered):
- Q₁ is at position n/4
- Q₂ (median) is at position n/2
- Q₃ is at position 3n/4
If the position is not a whole number, round up to the next data point (or interpolate, depending on the method specified).
Box Plots (Box and Whisker Diagrams)
A box plot displays:
- Minimum (or lowest non-outlier)
- Q₁ (left edge of box)
- Median (line inside box)
- Q₃ (right edge of box)
- Maximum (or highest non-outlier)
- Outliers plotted as individual points
Box plots are excellent for comparing distributions — you can see differences in location (median), spread (IQR), and skewness at a glance.
Outliers
An outlier is a data point that lies unusually far from the rest. Common definitions:
- Below Q₁ - 1.5 × IQR or above Q₃ + 1.5 × IQR
- More than 2 (or 3) standard deviations from the mean
Worked Example: Q₁ = 20, Q₃ = 35, IQR = 15.
Lower fence = 20 - 1.5(15) = 20 - 22.5 = -2.5
Upper fence = 35 + 1.5(15) = 35 + 22.5 = 57.5
Any data point below -2.5 or above 57.5 is an outlier.
Important: An outlier is not necessarily an error. It could represent genuine extreme behaviour (e.g., an unusually tall person). Decisions about whether to include or exclude outliers should be based on context.
Coding (Linear Transformation)
Coding simplifies calculations by transforming data. If the coding is y = (x - a) / b, then:
- Mean: x̄ = b × ȳ + a (reverse the coding)
- Standard deviation: σₓ = b × σᵧ (scaling only; the shift a does not affect spread)
Worked Example: Data is coded using y = (x - 100)/5. The coded data has mean ȳ = 3.2 and standard deviation σᵧ = 1.4.
Original mean: x̄ = 5(3.2) + 100 = 16 + 100 = 116
Original standard deviation: σₓ = 5(1.4) = 7
Histograms
In a histogram, the area of each bar represents frequency, not the height. The y-axis shows frequency density:
Frequency density = frequency / class width
This allows classes of unequal width to be compared fairly.
To find the frequency for a class: frequency = frequency density × class width.
Cumulative Frequency Diagrams
Plot the upper class boundary on the x-axis against the cumulative frequency on the y-axis. Join points with a smooth curve. Use the diagram to estimate:
- Median (read across from n/2)
- Quartiles (from n/4 and 3n/4)
- Percentiles and the IQR
Skewness
- Positive skew: tail extends to the right; mean > median > mode
- Negative skew: tail extends to the left; mean < median < mode
- Symmetric: mean ≈ median ≈ mode
Comparing Data Sets
When comparing distributions, always comment on:
1. A measure of location (e.g., "the median for Group A is higher, suggesting...")
2. A measure of spread (e.g., "the IQR for Group B is larger, indicating more variability...")
3. Context — relate your comparison to the real-world situation
Exam Tips
- For grouped data, always use midpoints — do not use class boundaries for the mean.
- When using the variance formula Σx²/n - x̄², keep extra decimal places throughout to avoid rounding errors.
- In coding questions, remember that addition/subtraction affects only the mean, while multiplication/division affects both mean and standard deviation.
- Box plot comparisons need two statistical comments (location and spread) plus a contextual sentence.
- In histogram questions, if class widths are unequal, you must use frequency density, not raw frequency.