Statistics is the branch of mathematics that deals with collecting, organising, analysing and interpreting numerical data. For AP TET Paper II, this topic carries consistent weightage because it connects mathematical skills with real-world applications—a key pedagogical objective for classes 6–8.
You must master three central tendencies (mean, median, mode), understand range as a measure of dispersion, and confidently interpret data presented in tables, bar graphs, pictographs and pie charts. Questions typically test both calculation skills and the ability to choose the appropriate measure for a given data set. Since this topic also appears in the pedagogy section, understanding *why* we teach statistics to children (data literacy, decision-making, pattern recognition) strengthens both content and methodology answers.
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Key Concepts
**Data** is a collection of facts or numbers gathered for analysis. Data can be *primary* (collected first-hand) or *secondary* (obtained from existing sources).
**Mean (Arithmetic Average)** is the sum of all observations divided by the number of observations. It uses every data point and is sensitive to extreme values (outliers).
**Median** is the middle value when data is arranged in ascending or descending order. It is unaffected by outliers and better represents "typical" value in skewed distributions.
**Mode** is the observation that occurs most frequently. A data set can be unimodal (one mode), bimodal (two modes), multimodal or have no mode if all values appear equally.
**Range** measures spread: Range = Highest value − Lowest value. A larger range indicates greater variability.
**Frequency** is the number of times a particular observation occurs. Frequency tables organise raw data for easier analysis.
**Grouped data** uses class intervals (e.g., 10–20, 20–30) when dealing with large data sets; ungrouped data lists individual values.
For basic data interpretation, always read axis labels, titles and units carefully before answering questions on graphs.
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Formulas / Key Facts
| Measure | Formula / Method | |---------|------------------| | **Mean** | Mean = (Sum of all observations) ÷ (Number of observations) | | **Mean (frequency)** | Mean = Σ(f × x) ÷ Σf, where f = frequency, x = observation | | **Median (odd n)** | Middle term = value at position (n + 1)/2 after arranging data | | **Median (even n)** | Average of values at positions n/2 and (n/2 + 1) | | **Mode** | Observation with highest frequency | | **Range** | Range = Maximum value − Minimum value |
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The marks obtained by 7 students in a mathematics test are: 15, 18, 22, 15, 20, 18, 15. What is the mode of this data?
Q2 · Statistics · MEDIUM
The monthly savings (in rupees) of 6 families are: 500, 600, 800, 700, 900, 500. Find the median savings.
Q3 · Statistics · EASY
The heights (in cm) of 5 students are: 140, 145, 150, 155, 160. What is the range of the heights?
Q4 · Statistics · MEDIUM
The mean of 5 numbers is 24. If four of the numbers are 20, 22, 26, and 28, what is the fifth number?
Q5 · Statistics · HARD
A teacher recorded the number of books read by 8 students in a month: 3, 5, 7, 5, 8, 5, 6, 9. Find the mean number of books read. If the mode is subtracted from the mean, what is the result?
1. Mean is best when data has no extreme values. 2. Median is preferred for skewed data (e.g., income distribution). 3. Mode is useful for categorical data (e.g., most popular shoe size). 4. Range alone does not reveal how data is distributed between extremes. 5. In a symmetric distribution, Mean ≈ Median ≈ Mode.
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Worked Examples
### Example 1: Finding Mean, Median, Mode and Range
**Data:** Marks of 9 students: 45, 55, 55, 60, 65, 70, 75, 80, 95
**Mean:** Sum = 45 + 55 + 55 + 60 + 65 + 70 + 75 + 80 + 95 = 600 Number of observations = 9 Mean = 600 ÷ 9 = **66.67** (approx.)
**Median:** Data is already in ascending order. n = 9 (odd). Position = (9 + 1)/2 = 5th term Median = **65**
### Example 3: Median for Even Number of Observations
**Data:** 12, 18, 22, 25, 30, 35
n = 6 (even) Positions: n/2 = 3rd term = 22; (n/2 + 1) = 4th term = 25 Median = (22 + 25) ÷ 2 = **23.5**
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Common Mistakes
1. **Forgetting to arrange data before finding median.** Wrong: Picking the middle position from unordered data. Fix: Always sort data in ascending or descending order first.
2. **Confusing mean formula for grouped vs ungrouped data.** Wrong: Adding frequencies directly instead of computing Σ(f × x). Fix: For frequency data, multiply each observation by its frequency, sum these products, then divide by total frequency.
3. **Assuming every data set has exactly one mode.** Wrong: Forcing a single mode answer when two values share the highest frequency. Fix: Recognise bimodal or "no mode" situations and state them clearly.
4. **Misreading graph scales in data interpretation.** Wrong: Reading 25 as 250 because the scale increment was overlooked. Fix: Check the axis scale and unit labels before reading values.
5. **Using range alone to judge data consistency.** Wrong: Concluding two data sets are equally spread because ranges match. Fix: Range ignores internal distribution; consider all measures together.
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Quick Reference
**Mean** = Total sum ÷ Count — sensitive to outliers.
**Median** = Middle value (arrange first!) — robust to outliers.
**Mode** = Most frequent value — useful for categorical data.
**Range** = Max − Min — simple measure of spread.
For **odd n**: Median at (n + 1)/2; for **even n**: average of two middle terms.
Always read graph **titles, axis labels and scale** before interpreting data.
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*Revise these formulas with small data sets daily. Practice reading at least one bar graph or pie chart question per session to build speed and accuracy for the exam.*