Statistics I — Statistics & Probability (16 lessons)

Subject: Core Mathematics | Target age: 15 | Context: Kenyan (use local data sources such as market prices, school records, rainfall, county statistics)

Specific learning outcomes

1. Identify and outline sub-sub-strands:
   • Collection of data
   • Frequency distribution table
   • Mean, mode and median (grouped and ungrouped)
   • Representation of data (histogram & frequency polygon)
   • Interpretation of data
2. Collect data from real-life sources (school, market, weather stations, county offices, household surveys).
3. Draw frequency distribution tables for grouped and ungrouped data.
4. Determine mean, mode and median for grouped and ungrouped data.
5. Represent data using histograms and frequency polygons.
6. Interpret data from histograms and frequency polygons.
7. Promote data collection, organisation and representation for informed decision making.

16-Lesson scheme (suggested)

1. Lesson 1: Introduction to statistics — types of data (qualitative/quantitative), variables.
2. Lesson 2: Sources of data — primary & secondary; designing simple questionnaires.
3. Lesson 3: Collecting data — classroom survey (scores, household size, market prices).
4. Lesson 4: Organising ungrouped data; tallying and simple lists.
5. Lesson 5: Constructing frequency distribution tables (ungrouped).
6. Lesson 6: Grouped data — choosing class intervals and widths.
7. Lesson 7: Frequency distribution tables (grouped) — examples & practice.
8. Lesson 8: Mean for ungrouped data (definition, worked examples).
9. Lesson 9: Mean for grouped data (midpoint method; examples).
10. Lesson 10: Median for ungrouped and grouped data (formulas & practice).
11. Lesson 11: Mode (ungrouped and grouped); modal class & formula.
12. Lesson 12: Drawing histograms — axis, bars, equal/unequal class widths.
13. Lesson 13: Drawing frequency polygons — midpoints and connecting lines.
14. Lesson 14: Interpreting histograms & frequency polygons — central tendency & spread.
15. Lesson 15: Project work — class/school survey, analysis & presentation.
16. Lesson 16: Revision, assessment and real-life decision-making discussions.

1. Collection of Data (how & where)

• Primary: classroom tests, pupil heights, household sizes, market prices (e.g., maize per Kg), daily temperature or rainfall readings from a local weather station.
• Secondary: Kenya National Bureau of Statistics (KNBS) summaries, county reports, school archives, past KCSE/KCPE summary tables.
• Practical tips: use simple question forms, sample fairly (e.g., all streams or classes), obtain permission for household work, record date and location.

2. Frequency distribution tables

Ungrouped data: list each distinct value with its frequency (good for small data sets).

Example (ungrouped): Test scores (10 pupils): 56, 70, 45, 70, 82, 56, 90, 45, 60, 76

Score	Frequency
45	2
56	2
60	1
70	2
76	1
82	1
90	1

Grouped data: combine values into class intervals (useful for large data sets).

Class interval	Frequency (f)	Midpoint (m)	f × m
40 – 49	2	44.5	89.0
50 – 59	2	54.5	109.0
60 – 69	3	64.5	193.5
70 – 79	2	74.5	149.0
80 – 89	1	84.5	84.5
Total	10		625.0

3. Measures of central tendency (worked examples)

Mean — ungrouped

Formula: mean = (Σx) / n. Using the 10 test scores above: Σx = 650, n = 10 so mean = 650 ÷ 10 = 65.

Median — ungrouped

Sort values: 45, 45, 56, 56, 60, 70, 70, 76, 82, 90. For even n=10, median = average of 5th and 6th values = (60 + 70) ÷ 2 = 65.

Mode — ungrouped

The value(s) that occur most often: 45, 56 and 70 each occur twice → multimodal.

Mean — grouped (midpoint method)

Compute Σ(f × m) and divide by N. From the grouped table above: Σ(f × m) = 625, N = 10 → mean = 625 ÷ 10 = 62.5 (approx).

Median — grouped (formula)

Use: Median = L + ((N/2 − cf) / f) × h
Where L = lower boundary of median class, cf = cumulative frequency before median class, f = frequency of median class, h = class width.

Example: N=10 → N/2=5. Cumulative frequencies: 2, 4, 7 → median class = 60–69. Take L = 59.5, cf = 4, f = 3, h = 10.
Median = 59.5 + ((5 − 4) / 3) × 10 = 59.5 + 3.333 = 62.833 (≈ 62.83).

Mode — grouped (formula)

Use: Mode = L + ((fm − f1) / (2fm − f1 − f2)) × h
Where fm = frequency of modal class, f1 = frequency before, f2 = frequency after.

For modal class 60–69: L=59.5, fm=3, f1=2, f2=2, h=10: Mode = 59.5 + ((3 − 2) / (6 − 2 − 2)) × 10 = 59.5 + 5 = 64.5.

4. Representing data — histogram & frequency polygon

Rules for histograms:

• Horizontal axis: class intervals; vertical axis: frequency (or frequency density if class widths differ).
• Bars touch each other; height = frequency (for equal widths).

Frequency polygon

Plot frequency against midpoints and join the points with straight lines. Extend to class boundaries at ends (optional) to close polygon.

5. Interpreting data (what to look for)

• Central tendency: compare mean, median, mode — are they close? Large differences may indicate skewness or outliers.
• Spread: range, variability of bars — many scattered small bars vs one tall bar.
• Shape: symmetric, positively skewed (tail to right), negatively skewed (tail to left), or uniform.
• Modal class: where most observations fall — useful for decisions like targeting interventions.
• Trends by local data: e.g., if many pupils score below passing mark, plan remedial lessons; if most vendors sell at similar prices, recommend bulk-buy strategies.

6. Practical classroom activities / suggested learning experiences

1. Class survey: collect test scores, daily travel time, household size or number of siblings. Organise into frequency tables and compute mean, median and mode.
2. Market study: in groups, collect prices of maize, beans, sukuma wiki from 5 local markets. Create grouped tables and draw histograms; discuss price ranges and recommendations for small traders.
3. Weather log: record daily rainfall or temperature for one month; make frequency distribution and a frequency polygon; interpret seasonal patterns.
4. Project: each group investigates one real-life question (e.g., typical pocket money in the class). Present results (tables, histogram, interpretation) to class or school management.
5. Use past KCSE/KNEC summary tables or county data for practice in interpreting larger datasets (teacher-led).

7. Assessment ideas

• Short quiz: compute mean, median, mode for small ungrouped and grouped datasets.
• Practical test: collect a small dataset, prepare frequency table, draw histogram/frequency polygon, and write a short interpretation (3–5 sentences).
• Project mark: accuracy of data collection, correct tables/graphs, clarity of interpretation, and usefulness of recommendations.

8. Classroom reminders & good practice

• Always label axes and give units (e.g., scores, KSh, mm of rainfall).
• Choose class widths sensibly for grouped data (equal widths usually easier).
• Be careful with continuous class boundaries (e.g., 59.5–69.5) when applying grouped formulas.
• Check for outliers and discuss whether to include or investigate them.
• Ethics: get permission for household work; anonymise personal data when presenting.

Use these notes to structure the 16 lessons, adapting local examples (county statistics, school data, market prices) to make learning relevant for Kenyan 15-year-olds. Encourage students to collect, represent and interpret data to support decisions (school projects, community recommendations).