Mathematics Form 4 | Statistics (2) | Masomotele Free Learning

📋 Learning Outcomes

By the end of this lesson, you should be able to:

Calculate measures of central tendency for grouped data
Calculate the standard deviation and variance
Interpret standard deviation in context
Draw and interpret histograms and frequency polygons
Calculate quartiles and interquartile range
Draw and interpret cumulative frequency curves (ogives)
Apply statistics to real-life data analysis

🔍 Key Inquiry Question: How do we summarise and compare large sets of data meaningfully?

📊 Mean, Median, Mode for Grouped Data

Mean from Grouped Data

x̄ = Σfx / Σf

where x = midpoint of class, f = frequency

Example:

Class	f	Midpoint x	fx
10-19	5	14.5	72.5
20-29	8	24.5	196
30-39	7	34.5	241.5

Mean = (72.5+196+241.5)/(5+8+7) = 510/20 = 25.5

Modal Class

The class with the highest frequency. In the example above: modal class = 20-29

📏 Standard Deviation

Standard Deviation Formula

σ = √[Σf(x−x̄)² / Σf]

Or: σ = √[Σfx²/Σf − (x̄)²]

Standard deviation measures how spread out the data is from the mean. Small σ = data clustered near mean. Large σ = data widely spread.

Example (ungrouped):

Data: 2, 4, 4, 4, 5, 5, 7, 9. Mean = 5

Deviations²: 9,1,1,1,0,0,4,16. Sum = 32

σ = √(32/8) = √4 = 2

Variance

Variance = σ² = Σf(x−x̄)² / Σf

📈 Quartiles and IQR

Quartiles

Divide ordered data into four equal parts.

Quartile	Position	Meaning
Q1 (Lower)	¼ of way	25th percentile
Q2 (Median)	½ of way	50th percentile
Q3 (Upper)	¾ of way	75th percentile

IQR = Q3 − Q1

Interquartile Range — measures spread of middle 50% of data

Example:

Data: 3, 5, 7, 9, 11, 13, 15, 17, 19. n=9

Q1 = 2nd+3rd avg = (5+7)/2? No — Q1 position = (9+1)/4 = 2.5 → Q1 = (5+7)/2 = 6

Q3 position = 3(9+1)/4 = 7.5 → Q3 = (15+17)/2 = 16

IQR = 16−6 = 10

📉 Cumulative Frequency (Ogive)

Building a Cumulative Frequency Table

Class	Freq	Cum. Freq
0-9	4	4
10-19	8	12
20-29	12	24
30-39	6	30

Reading from Ogive

Median → read off at cumulative frequency = n/2
Q1 → read off at n/4
Q3 → read off at 3n/4
Percentiles → read off at required percentage of n

📊 Histograms and Frequency Polygons

Histograms

Used for continuous grouped data. The area of each bar represents frequency.

Frequency Density = Frequency / Class Width (for unequal class widths)

Frequency Polygon

Formed by joining the midpoints of the tops of histogram bars.

Uses of frequency polygon:

Compare two distributions on the same graph
Show the shape of distribution (symmetric, skewed)
Identify modal class visually

🔢 Comparing Distributions

Using Mean and SD to Compare

Example:

Class A: mean = 65, σ = 5

Class B: mean = 65, σ = 15

Both classes have same average but Class B has more variation — scores more spread out. Class A performed more consistently.

Coefficient of Variation (CV):

CV = (σ/x̄) × 100% — compares variability between datasets with different means

Lower CV → more consistent data

💡 Real-Life Applications

Quality Control

Example:

A factory produces bolts. Target diameter = 10mm, σ = 0.1mm. A bolt is acceptable if within 2σ of mean (9.8mm to 10.2mm). This uses the concept of standard deviation in manufacturing.

Finance — Risk Analysis

Example:

Investment A: mean return 12%, σ = 2%

Investment B: mean return 12%, σ = 8%

Investment A is less risky (lower standard deviation = less variability in returns).

✏️ Practice Problems

Problem 1

Calculate mean and standard deviation of: 5, 8, 10, 12, 15.

Solution:

Mean = 50/5 = 10

Deviations: −5,−2,0,2,5. Squared: 25,4,0,4,25. Sum = 58

σ = √(58/5) = √11.6 ≈ 3.41

Problem 2

Find Q1, Q3, IQR for: 12, 15, 18, 20, 22, 25, 28, 30, 35.

Solution:

n=9. Q1 = average of 2nd and 3rd = (15+18)/2 = 16.5

Q3 = average of 7th and 8th = (28+30)/2 = 29

IQR = 29−16.5 = 12.5

📝 Quiz — Statistics (2)

Part A: Fill in the Blanks (5 marks)

1. The sum of all values divided by the number of values gives the .

2. The middle value in ordered data is the .

3. The most frequently occurring value is the .

4. The square root of variance is called .

5. σ² is the symbol for .

Part B: Short Answer (10 marks)

6. (4 marks) Calculate the mean and standard deviation of the grouped data: 10-19 (f=3), 20-29 (f=7), 30-39 (f=5), 40-49 (f=5).

7. (3 marks) Data: 4, 7, 9, 12, 14, 16, 19, 21. Find Q1, Q3, and IQR.

8. (3 marks) Two factories A and B produce rods. Factory A: mean length=50cm, σ=0.5cm. Factory B: mean=50cm, σ=2cm. Which factory is more consistent and why?