The Logic of Dispersion#
In Grade 10 you learned to summarise data with the mean, median, and mode — measures of central tendency (where the data clusters). But knowing the centre isn’t enough. Two classes can both average 60% on a test, yet one class has everyone between 55% and 65% while the other has marks scattered from 20% to 100%.
Dispersion measures tell you how spread out the data is. Grade 11 introduces two precise measures: the variance and the standard deviation.
1. Why Not Just Use the Range?#
The range ($\text{max} - \text{min}$) is a simple measure of spread, but it only uses two data points. One extreme value changes the range dramatically while telling you nothing about what’s happening in between.
We need a measure that uses every data point. The natural idea: measure how far each point is from the mean, then average those distances.
The Problem with Simple Averages of Deviations#
If you calculate $x - \bar{x}$ for every data point and add them up, you always get zero — the positive deviations cancel the negative ones perfectly. (This is a mathematical property of the mean.)
| $x$ | $x - \bar{x}$ |
|---|---|
| 4 | $-4$ |
| 7 | $-1$ |
| 8 | $0$ |
| 10 | $+2$ |
| 11 | $+3$ |
| Sum | 0 |
The fix: Square every deviation first. Squaring makes everything positive and also penalises large deviations more heavily (which is what we want — a point 10 units away is more concerning than two points each 5 units away).
2. Variance ($\sigma^2$)#
The variance is the mean of the squared deviations:
$$\boxed{\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}}$$| Symbol | Meaning |
|---|---|
| $x_i$ | Each individual data value |
| $\bar{x}$ | The mean of all data values |
| $n$ | The number of data values |
| $\sigma^2$ | The variance |
The variance tells you the average squared distance from the mean. Its unit is the square of the original unit (e.g., if data is in cm, variance is in cm²).
3. Standard Deviation ($\sigma$)#
To get back to the original units, take the square root of the variance:
$$\boxed{\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}}$$What it means in plain English: On average, each data point sits about $\sigma$ units away from the mean.
Interpreting Standard Deviation#
| $\sigma$ value | What it means |
|---|---|
| Small $\sigma$ | Data is tightly clustered around the mean — consistent |
| Large $\sigma$ | Data is widely spread — variable, less predictable |
| $\sigma = 0$ | Every data value is identical (no spread at all) |
The 68% Rule: In roughly symmetric data, about 68% of the data falls within one standard deviation of the mean: $[\bar{x} - \sigma;\; \bar{x} + \sigma]$.
4. Worked Example — Standard Deviation by Hand#
Data: $4;\; 7;\; 8;\; 10;\; 11$ ($n = 5$)
Step 1 — Calculate the mean:
$$\bar{x} = \frac{4 + 7 + 8 + 10 + 11}{5} = \frac{40}{5} = 8$$Step 2 — Build the deviation table:
| $x_i$ | $x_i - \bar{x}$ | $(x_i - \bar{x})^2$ |
|---|---|---|
| 4 | $-4$ | 16 |
| 7 | $-1$ | 1 |
| 8 | $0$ | 0 |
| 10 | $+2$ | 4 |
| 11 | $+3$ | 9 |
| Sum | 0 ✓ | 30 |
Check: The deviations column must sum to 0. If it doesn’t, your mean is wrong.
Step 3 — Variance:
$$\sigma^2 = \frac{30}{5} = 6$$Step 4 — Standard deviation:
$$\sigma = \sqrt{6} \approx 2.45$$Interpretation: On average, each data point is about 2.45 units away from the mean of 8.
5. Standard Deviation from a Frequency Table#
When data is given with frequencies, the formula adjusts to account for repeated values:
$$\boxed{\sigma = \sqrt{\frac{\sum f_i(x_i - \bar{x})^2}{\sum f_i}}}$$The mean is: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$
Worked Example — Frequency Table#
Test scores for 20 learners:
| Score ($x_i$) | Frequency ($f_i$) | $f_i \cdot x_i$ |
|---|---|---|
| 3 | 2 | 6 |
| 5 | 5 | 25 |
| 7 | 8 | 56 |
| 9 | 4 | 36 |
| 10 | 1 | 10 |
| Total | 20 | 133 |
Step 1 — Mean: $\bar{x} = \frac{133}{20} = 6.65$
Step 2 — Squared deviations × frequency:
| $x_i$ | $f_i$ | $x_i - \bar{x}$ | $(x_i - \bar{x})^2$ | $f_i(x_i - \bar{x})^2$ |
|---|---|---|---|---|
| 3 | 2 | $-3.65$ | $13.3225$ | $26.645$ |
| 5 | 5 | $-1.65$ | $2.7225$ | $13.6125$ |
| 7 | 8 | $0.35$ | $0.1225$ | $0.98$ |
| 9 | 4 | $2.35$ | $5.5225$ | $22.09$ |
| 10 | 1 | $3.35$ | $11.2225$ | $11.2225$ |
| Total | 20 | 74.55 |
Step 3 — Variance: $\sigma^2 = \frac{74.55}{20} = 3.7275$
Step 4 — Standard deviation: $\sigma = \sqrt{3.7275} \approx 1.93$
6. Grouped Data (Class Intervals)#
When data is given in intervals (e.g., 10–20, 20–30, …), you don’t know the exact values. Use the midpoint of each class as $x_i$:
$$\text{midpoint} = \frac{\text{lower boundary} + \text{upper boundary}}{2}$$Worked Example — Grouped Data#
Heights of 50 Grade 11 learners:
| Height (cm) | Midpoint ($x_i$) | Frequency ($f_i$) | $f_i \cdot x_i$ |
|---|---|---|---|
| 140–150 | 145 | 4 | 580 |
| 150–160 | 155 | 12 | 1860 |
| 160–170 | 165 | 20 | 3300 |
| 170–180 | 175 | 10 | 1750 |
| 180–190 | 185 | 4 | 740 |
| Total | 50 | 8230 |
Mean: $\bar{x} = \frac{8230}{50} = 164.6$ cm
Now calculate $\sum f_i(x_i - \bar{x})^2$:
| $x_i$ | $f_i$ | $(x_i - 164.6)^2$ | $f_i(x_i - 164.6)^2$ |
|---|---|---|---|
| 145 | 4 | $384.16$ | $1536.64$ |
| 155 | 12 | $92.16$ | $1105.92$ |
| 165 | 20 | $0.16$ | $3.20$ |
| 175 | 10 | $108.16$ | $1081.60$ |
| 185 | 4 | $416.16$ | $1664.64$ |
| Total | 50 | 5392.00 |
Interpretation: The heights are spread about 10.4 cm either side of the mean on average.
⚠️ Grouped data gives an approximation — we assumed all values in each class are at the midpoint. The true $\sigma$ may differ slightly.
7. Histograms, Frequency Polygons & Ogives#
Histograms#
- Class intervals on the x-axis, frequency on the y-axis
- Bars touch (no gaps) — the data is continuous
- The tallest bar = the modal class (the interval with the highest frequency)
- Bar width equals the class width
Frequency Polygons#
- Plot the midpoint of each class against its frequency
- Connect the points with straight lines
- Extend to the x-axis one class interval before and after the data (so the polygon closes)
Ogives (Cumulative Frequency Curves)#
An ogive shows the running total of frequencies. It answers the question: “How many data values are less than or equal to this boundary?”
How to draw an ogive:
- Create a cumulative frequency column (running total)
- Plot cumulative frequency against the upper boundary of each class
- Start with the point (lower boundary of first class, 0)
- Connect with a smooth S-shaped curve
Worked Example — Reading an Ogive#
Using the height data from Section 6:
| Upper boundary | Cumulative frequency |
|---|---|
| 140 | 0 |
| 150 | 4 |
| 160 | 16 |
| 170 | 36 |
| 180 | 46 |
| 190 | 50 |
Reading values from the ogive:
- Median ($Q_2$): at $\frac{50}{2} = 25$ → read across from 25 on the y-axis to the curve → read down to get approximately 164 cm
- $Q_1$: at $\frac{50}{4} = 12.5$ → approximately 158 cm
- $Q_3$: at $\frac{3 \times 50}{4} = 37.5$ → approximately 172 cm
- IQR = $172 - 158 = 14$ cm
Exam tip: When a question says “use the ogive to estimate the median,” draw horizontal and vertical dashed lines on the graph and label the coordinates. Show your working on the graph — marks are awarded for this.
8. Comparing Two Datasets#
A very common exam question: “Compare the performance of Class A and Class B.”
The Strategy#
| Compare… | Using… |
|---|---|
| Centre (who did better overall?) | Mean or median |
| Spread (who was more consistent?) | Standard deviation or IQR |
Worked Example — Comparing Classes#
Class A: $\bar{x} = 62$, $\sigma = 8$
Class B: $\bar{x} = 62$, $\sigma = 15$
Centre: Both classes have the same mean (62%), so their overall performance was the same.
Spread: Class A has a much lower standard deviation ($\sigma = 8$ vs $\sigma = 15$), so Class A’s marks were more consistent (clustered closer to the mean). Class B had a wider range of marks — some very high, some very low.
Another Example — Different Means#
Team X: $\bar{x} = 45$, $\sigma = 5$
Team Y: $\bar{x} = 52$, $\sigma = 12$
Centre: Team Y scored higher on average (52 vs 45).
Spread: Team X was more consistent ($\sigma = 5$ vs $\sigma = 12$). Team Y had higher scores overall but with much more variation between players.
Exam phrasing: “The data for [X] is more consistent/spread out than [Y] because the standard deviation is lower/higher.”
9. Symmetric vs Skewed Data#
The shape of the data distribution tells you about the relationship between mean and median.
| Shape | Mean vs Median | Box plot clue | Histogram clue |
|---|---|---|---|
| Symmetric | Mean ≈ Median | Median centred in box, whiskers roughly equal | Bell-shaped, mirror image |
| Positively skewed (right) | Mean > Median | Median closer to $Q_1$, long right whisker | Tail stretches to the right |
| Negatively skewed (left) | Mean < Median | Median closer to $Q_3$, long left whisker | Tail stretches to the left |
Why the mean shifts: The mean is pulled toward extreme values (outliers/tail), while the median stays in the middle of the ranked data. So in positively skewed data, a few very high values drag the mean above the median.
Which Measure to Use?#
| Data shape | Best centre measure | Best spread measure |
|---|---|---|
| Symmetric | Mean | Standard deviation |
| Skewed | Median | IQR |
10. Outliers#
An outlier is a data value that is unusually far from the rest. The formal test:
$$\boxed{x < Q_1 - 1.5 \times IQR \quad \text{or} \quad x > Q_3 + 1.5 \times IQR}$$Where $IQR = Q_3 - Q_1$.
Worked Example — Detecting Outliers#
Data: $2;\; 15;\; 18;\; 20;\; 22;\; 23;\; 25;\; 27;\; 28;\; 30;\; 65$
$Q_1 = 18$, $Q_3 = 28$, $IQR = 10$
Lower fence: $18 - 1.5(10) = 3$
Upper fence: $28 + 1.5(10) = 43$
Any value below 3 or above 43 is an outlier.
- $x = 2$: below 3 → outlier ✓
- $x = 65$: above 43 → outlier ✓
Impact of Outliers on Statistics#
| Measure | Affected by outliers? |
|---|---|
| Mean | Yes — dragged toward the outlier |
| Median | No — only depends on the middle value(s) |
| Standard deviation | Yes — outliers inflate squared deviations |
| IQR | No — only depends on $Q_1$ and $Q_3$ |
| Range | Yes — completely determined by extremes |
Exam tip: If a question asks “which measure of central tendency is more appropriate?”, check for skewness or outliers. If present, the median is better than the mean.
🚨 Common Mistakes#
| Mistake | Why it’s wrong | Fix |
|---|---|---|
| Using $s_x$ instead of $\sigma_x$ on the calculator | $s_x$ is the sample standard deviation (divides by $n-1$); school maths uses $\sigma_x$ (divides by $n$) | Always select $\sigma_x$ in STAT mode |
| Forgetting to multiply by frequency | In a frequency table, each $(x - \bar{x})^2$ must be weighted by its frequency | Always include the $f_i$ column in your table |
| Forgetting the square root | $\sigma^2$ is the variance; $\sigma$ is the standard deviation | The last step is always $\sigma = \sqrt{\sigma^2}$ |
| Using midpoints as exact values | Grouped data uses midpoints as estimates, not exact values | Acknowledge “estimated” in your answer |
| Reading ogive at the wrong axis | The median is at cumulative frequency $\frac{n}{2}$, not at $x = \frac{n}{2}$ | Start on the y-axis, go across to the curve, then down to the x-axis |
| Confusing skewness direction | “Positively skewed” means the tail goes to the right (positive direction) | Remember: the name matches the direction of the tail, not the peak |
| Stating “data is consistent” without referencing $\sigma$ | You must mention the standard deviation value to justify a consistency claim | “The data is more consistent because $\sigma = 5$ is lower than $\sigma = 12$” |
💡 Pro Tips for Exams#
1. Calculator Method#
Use STAT mode: enter all data → select 1-VAR stats → read $\bar{x}$ and $\sigma_x$ directly. The by-hand method is for understanding and for “show your working” questions; the calculator is for speed.
2. The “Deviation Sum = 0” Check#
After calculating all your deviations ($x_i - \bar{x}$), add them up. If the sum isn’t 0, your mean is wrong. Fix it before continuing.
3. Comparing Datasets — The Two-Sentence Template#
When asked to compare two datasets, always make two statements:
- Centre: “Class A performed better/worse on average because [mean/median] = …”
- Spread: “Class A was more/less consistent because [σ/IQR] = …”
This guarantees full marks on comparison questions.
4. Reading Ogives Accurately#
Always draw dashed lines on the ogive when reading values. Start from the y-axis (cumulative frequency), draw a horizontal line to the curve, then a vertical line down to the x-axis. Label both coordinates.
🔗 Related Grade 11 topics:
- Combined Events & Probability — contingency tables link statistics and probability
- Quadratic Equations — the variance formula involves squaring, a key algebraic skill