What are Measures of Central Tendency?

When we collect a large set of data — marks, heights, salaries, weights — we need a single value that best represents the entire dataset. This representative value is called a measure of central tendency. It tells us where the "centre" of the data lies. In Class 9 Statistics, three such measures are studied:

1. Arithmetic Mean — Formulas and Concepts

Mean of Raw (Ungrouped) Data

When observations are listed directly without frequencies, the mean is simply the total divided by the count.

Example from the textbook: Data: 4, 3, 5, 5, 4, 9, 7, 7, 8, 7, 4, 3, 7, 5, 8, 8, 8, 7, 4, 7 (20 observations)

Mean of Frequency Distribution

When the same values repeat, we use a frequency table. Each value xᵢ appears fᵢ times, so we multiply them to get the weighted sum:

Using the same data grouped by frequency: values 3,4,5,7,8,9 appear 2,4,3,6,4,1 times respectively.

Question 1 — Mean Weight of Parcels

Question 2 — Mean Number of Children per Family

Question 3 — Finding the Unknown Frequency 'K'

Question 4 — Average Village Population (Census 2011)

Question 5 — AFLATOUN School Savings Programme (Hyderabad)

2. Median — Finding the Middle Value

The median is the middle observation of a dataset arranged in ascending (or descending) order. It divides the data into two equal halves — one half has values below the median and the other half has values above it.

Median Example 1 (Odd n):

Data: 75, 21, 56, 36, 81, 05, 42 → Arranged: 5, 21, 36, 42, 56, 75, 81

Median Example 2 (Even n):

Data: 75, 21, 56, 36, 81, 05, 42, 68 → Arranged: 5, 21, 36, 42, 56, 68, 75, 81

Median of a Frequency Distribution

Build a cumulative frequency (cf) column. The median is the observation whose cumulative frequency first reaches or exceeds N/2, where N = Σfᵢ.

Worked example (wages): N = 100, N/2 = 50. The cf first reaches 52 at wage ₹8500.

Question 6 — Comparing Heights of Boys and Girls

3. Mode — The Most Frequent Value

The mode is the observation that occurs most often in a dataset. In a frequency table, the mode is simply the value with the highest frequency.

• Unimodal data — one value has the highest frequency (e.g., mode = 7 appearing 5 times)
• Bimodal data — two values tie for the highest frequency (e.g., both 4 and 6 appear 5 times each)
• Trimodal data — three values have the same highest frequency
• No mode — all values appear equally often; no single observation dominates

Question 7 — Centuries Scored by Cricketers (Mean + Median + Mode)

Question 8 — Sweet Packets on New Year's Day (Mean + Median + Mode)

Question 9 — Finding an Unknown Weight Using Mean

Question 10 — School Donations to Orphanage (Mean + Median + Mode)

Question 11 — Four Unknown Numbers Using Mean Equations

Quick Summary — All Questions at a Glance

Common Mistakes to Avoid in Board Exams

• Forgetting to arrange data before finding median: The median formula only works on sorted data. Always arrange values in ascending order first — skipping this step gives a completely wrong answer.
• Using N/2 instead of (N+1)/2 for odd n in raw data: For ungrouped data with odd n, use (n+1)/2. The N/2 shortcut applies to finding the median class in a frequency distribution, not raw data.
• Confusing mode with median: Mode = most frequent value. Median = middle value. These are different. In Q10, both happen to be ₹10, but that is a coincidence.
• Not writing units in the answer: If xᵢ is in thousands (Q4), remember to convert: 17.7 thousand = 17,700 persons. Always state units clearly.
• In Q3 type problems — not forming the equation first: Many students substitute wrongly. Always write Mean × Σf = Σfx and then substitute the given mean before expanding.

What This Exercise Prepares You For

The concepts in Exercise 9.2 are the starting point for all statistical analysis you will encounter in higher classes. The weighted mean formula (Σfᵢxᵢ / Σfᵢ) reappears in Class 10 Statistics Exercise 14.1, where it is extended to grouped data with class intervals using the direct method, assumed mean method, and step-deviation method.

The cumulative frequency technique used for median in this exercise is also the foundation for constructing ogives (cumulative frequency curves) in Class 10 Exercise 14.4. Bahadurpura's high mean in Q5 vs. other mandals is a classic example of why median can sometimes better represent a skewed dataset — a concept explored in detail in Class 11 and beyond.

For Telangana and Andhra Pradesh board examinations, Questions 3 (find K), 7 and 8 (all three measures), and 11 (chained mean) are high-frequency question types. Mastering these fully earns you 4–5 marks per problem in the annual examination.