Statistics and Data Presentation

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Statistics and Data Presentation — Quick Facts

Measures of Central Tendency:

Measure	Formula (Ungrouped)	Best Used When
Mean	$\bar{x} = \frac{\sum x_i}{n}$	Symmetric data, no outliers
Median	Middle value when sorted	Skewed data, outliers present
Mode	Most frequent value	Categorical data

Grouped Data Mean: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$

where $x_i$ is the class midpoint and $f_i$ is the frequency.

Dispersion Measures:

• Range = Highest − Lowest
• Variance = $\sigma^2 = \frac{\sum f_i(x_i - \bar{x})^2}{\sum f_i}$
• Standard Deviation = $\sigma = \sqrt{\text{variance}}$

Data Presentation Types:

• Bar chart — categorical data comparison
• Histogram — continuous data with equal class widths
• Frequency polygon — line graph of frequencies
• Pie chart — proportions of a whole

⚡ JAMB Exam Tip: In histogram questions, area = frequency. If class widths are unequal, use frequency density = frequency/class width.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Statistics and Data Presentation — Study Guide

Finding the Mean (Grouped Data):

Class Interval	Midpoint $x_i$	Frequency $f_i$	$f_i x_i$
0–9	4.5	3	13.5
10–19	14.5	7	101.5
20–29	24.5	12	294.0
30–39	34.5	8	276.0
40–49	44.5	2	89.0
Total		32	774.0

$\bar{x} = \frac{774.0}{32} = 24.19$

Finding the Median (Grouped Data):

Formula: $\text{Median} = L + \left(\frac{\frac{n}{2} - c}{f}\right) \times w$

Where:

• $L$ = lower boundary of median class
• $n$ = total frequency
• $c$ = cumulative frequency before median class
• $f$ = frequency of median class
• $w$ = class width

Example: Find median from data with $n = 50$, median class = 20–29 (cumulative before = 17, $f = 12$, $w = 10$)

$\text{Median} = 20 + \left(\frac{25 - 17}{12}\right) \times 10 = 20 + 6.67 = 26.67$

Measures of Position:

• Quartiles: $Q_1 = \frac{n+1}{4}$th term, $Q_3 = \frac{3(n+1)}{4}$th term
• Deciles: $D_k = \frac{k(n+1)}{10}$th term
• Percentiles: $P_k = \frac{k(n+1)}{100}$th term

⚡ Common Student Mistake: Confusing the median class in grouped data. Always identify the class containing the $\frac{n}{2}$th observation first.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Statistics and Data Presentation — Comprehensive Notes

Variance and Standard Deviation (Grouped Data):

Step 1: Calculate mean $\bar{x}$ Step 2: Calculate deviation from mean for each class midpoint: $(x_i - \bar{x})$ Step 3: Square deviations: $(x_i - \bar{x})^2$ Step 4: Multiply by frequency: $f_i(x_i - \bar{x})^2$ Step 5: Sum and divide by $\sum f_i$ Step 6: Take square root for standard deviation

Shortcut Formula for Variance: $$\sigma^2 = \frac{\sum f_i x_i^2}{\sum f_i} - \bar{x}^2$$

Example Calculation:

$x_i$	$f_i$	$f_i x_i$	$f_i x_i^2$
15	4	60	900
25	6	150	3750
35	8	280	9800
45	2	90	4050
Total	20	580	18500

$\bar{x} = 580/20 = 29$ $\sigma^2 = \frac{18500}{20} - 29^2 = 925 - 841 = 84$ $\sigma = \sqrt{84} = 9.17$

Skewness:

Pearson’s Coefficient of Skewness: $$S_k = \frac{3(\bar{x} - \text{Mode})}{\sigma}$$

• $S_k > 0$: Positively skewed (mean > median > mode)
• $S_k < 0$: Negatively skewed
• $S_k = 0$: Symmetrical

Cumulative Frequency (Ogive):

To draw an ogive:

1. Use upper class boundaries on x-axis
2. Use cumulative frequency on y-axis
3. Plot points and join with a smooth curve
4. Read median at $n/2$ on y-axis

Ogive for Median: 50th percentile from ogive = $\frac{n}{2}$ on y-axis, project down to x-axis.

Box-and-Whisker Plot:

• Lower whisker = $Q_1 - 1.5 \times \text{IQR}$
• Lower edge of box = $Q_1$
• Line inside box = Median ($Q_2$)
• Upper edge of box = $Q_3$
• Upper whisker = $Q_3 + 1.5 \times \text{IQR}$
• Outliers plotted beyond whiskers

Data Presentation Guidelines:

1. Histogram: No gaps between bars (continuous data), frequency on y-axis
2. Bar Chart: Gaps between bars (categorical data), frequency on y-axis
3. Frequency Polygon: Points plotted at class midpoints, connected
4. Pie Chart: Sectors proportional to frequencies; angle = $\frac{f_i}{\sum f_i} \times 360°$

⚡ Exam Pattern (JAMB 2015-2024):

• 2015: Grouped data mean with class midpoints
• 2018: Finding median from ogive
• 2020: Variance calculation using shortcut formula
• 2023: Histogram with unequal class widths (frequency density)
• 2024: Box-and-whisker plot interpretation

Real-World Nigeria Context Example:

In a JAMB mock exam, 500 students scored the following in Mathematics:

Score Range	Frequency
0–19	45
20–39	120
40–59	185
60–79	110
80–100	40

Find mean, median, and interpret the distribution shape.

Solution approach: Calculate class midpoints (9.5, 29.5, 49.5, 69.5, 89.5), multiply by frequencies, sum, divide by 500 to get mean ≈ 48.3. For median class, $n/2 = 250$, which falls in 40–59 class.