Introduction — Central Tendency
Arithmetic mean, median and mode using the deviation method.
Chapter 7: Frequency Distribution Tables and Graphs — Introduction
Chapter 7 of Class 8 Mathematics introduces students to one of the most practically useful branches of mathematics — Statistics. Whether you are reading a news report about exam results, a cricket score analysis, or rainfall data in your district, statistics is the tool that makes raw numbers meaningful. This chapter is part of the CBSE, Telangana, and Andhra Pradesh Class 8 syllabi and lays the groundwork for the more advanced statistical concepts you will encounter in Class 9 and Class 10.
The Introduction lesson focuses on three foundational ideas: Arithmetic Mean, Median, and Mode — collectively called the Measures of Central Tendency. Each one summarises an entire set of data with a single representative value, but in a different way. Understanding when to use which measure is the core skill this chapter builds.
The numerical average. Add all values and divide by how many there are.
The middle value when data is sorted. Unaffected by extreme values.
The most frequently occurring value. A dataset can have more than one mode.
What is Arithmetic Mean?
Measure of Central Tendency — 1 of 3The arithmetic mean — also simply called the mean or average — is calculated by adding up all the values in a dataset and then dividing that sum by the total number of values. It gives a single value that "represents" the entire dataset.
Arithmetic Mean (x̄) = Sum of all observations / Number of observations
If the observations in a dataset are x₁, x₂, x₃, … xₙ, then:
x̄ = (x₁ + x₂ + x₃ + … + xₙ) / n = Σxᵢ / n
Here, the symbol Σ (sigma) means "sum of", and n is the total count of observations. The mean is a unique value — no dataset can have two different arithmetic means — and it takes into account both the value and the count of every observation.
Worked Examples — Arithmetic Mean
Given data (daily sales in ₹):
| Day | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
|---|---|---|---|---|---|---|
| Sales (₹) | 5,000 | 4,200 | 4,800 | 6,600 | 5,400 | 5,200 |
Solution:
- Sum of all observations = 5000 + 4200 + 4800 + 6600 + 5400 + 5200 = ₹31,200
- Number of observations (n) = 6
- Arithmetic Mean = 31200 ÷ 6 = ₹5,200
Mean = 31,200 / 6 = ₹5,200
So the trader's average daily sales for the week were ₹5,200.
Here we use the reverse relationship. If you know the mean and the number of observations, you can always recover the total.
Sum of observations = Mean × Number of observations
- Mean = ₹5,200
- Number of days = 6
- Total sales = 5200 × 6 = ₹31,200
Key Properties of Arithmetic Mean
The arithmetic mean has several important properties that make it predictable and easy to work with. These properties are directly tested in CBSE, Telangana, and AP board exams.
- Representative value: The mean is a single number that fairly represents all the observations in a dataset.
- Uniqueness: Every dataset has exactly one arithmetic mean — it is a unique value.
- Depends on every observation: Unlike mode or median, the mean takes into account the value of each and every data point. Changing even one value changes the mean.
- Addition / Subtraction property: If a constant number is added to (or subtracted from) every observation, the mean also increases (or decreases) by that same constant.
- Multiplication / Division property: If every observation is multiplied (or divided) by a constant, the mean is also multiplied (or divided) by that same constant.
If we add 4 to each → new data: 11, 14, 19, 25, 31. New mean = 100 / 5 = 20 = 16 + 4. ✓
If we multiply each by 3 → new data: 21, 30, 45, 63, 81. New mean = 240 / 5 = 48 = 16 × 3. ✓
| Operation on Data | Original Mean | Effect on Mean | New Mean |
|---|---|---|---|
| Add 4 to each value | 16 | Mean also adds 4 | 20 |
| Subtract 4 from each value | 16 | Mean also subtracts 4 | 12 |
| Multiply each value by 3 | 16 | Mean also multiplies by 3 | 48 |
| Divide each value by 2 | 16 | Mean also divides by 2 | 8 |
What is Median?
Measure of Central Tendency — 2 of 3The median is the middle value of a dataset when the values are arranged in ascending or descending order. It divides the dataset exactly in half: 50% of values lie below the median and 50% lie above it. The median is especially useful when a dataset contains extreme values (outliers) that would distort the mean.
The formula to find the median depends on whether the number of observations n is odd or even:
If n is ODD: Median = value at position (n + 1) / 2
If n is EVEN: Median = average of values at positions n/2 and (n/2 + 1)
- Arrange in ascending order: 11, 12, 15, 19, 21
- Count observations: n = 5 (odd)
- Median position: (5 + 1) / 2 = 3rd observation
- Median = 15
- Arrange in ascending order: 11, 12, 15, 17, 19, 21
- Count observations: n = 6 (even)
- Median positions: n/2 = 3rd and (n/2 + 1) = 4th observations
- Values at 3rd and 4th positions: 15 and 17
- Median = (15 + 17) / 2 = 32 / 2 = 16
Median = (15 + 17) / 2 = 16
What is Mode?
Measure of Central Tendency — 3 of 3The mode is the value that appears most frequently in a dataset. Unlike mean or median, the mode requires no calculation — you simply identify which value repeats the most. Mode is particularly useful for categorical data (like the most popular colour, shoe size, or subject chosen by students).
| Dataset | Observation | Mode |
|---|---|---|
| 10, 8, 6, 8, 5, 7, 7, 8 | 8 appears 3 times — most frequent | Mode = 8 (Unimodal) |
| 7, 6, 5, 6, 7, 4, 6, 7 | Both 6 and 7 appear 3 times each | Mode = 6 and 7 (Bimodal) |
| 3, 5, 6, 7, 9, 11 | Every value appears exactly once | No mode |
A dataset can be unimodal (one mode), bimodal (two modes), multimodal (more than two modes), or have no mode at all if every value appears the same number of times.
Mean, Median, and Mode — Side-by-Side Comparison
| Feature | Arithmetic Mean | Median | Mode |
|---|---|---|---|
| Definition | Sum ÷ count | Middle value of sorted data | Most frequent value |
| Data must be sorted? | No | Yes | No |
| Affected by extreme values? | Yes | No | No |
| Can have more than one? | No (unique) | No (unique) | Yes (bimodal, multimodal) |
| Can be non-existent? | No | No | Yes (if all values appear equally) |
| Best used for… | Evenly spread numeric data | Data with outliers or skewed distribution | Categorical data / most popular item |
Common Mistakes to Avoid
- Not sorting data before finding median: The median formula only works on sorted (ascending or descending) data. Applying it to unsorted data gives a completely wrong answer.
- Confusing "middle position" with "middle value": The median formula gives you the position of the middle value — you must then look up what value sits at that position.
- Assuming there is always a mode: If every value in the dataset appears the same number of times, the data has no mode — this is a valid answer.
- Using the wrong formula for even/odd n: Always check whether n is odd or even before applying the median formula.
- Calculating mean instead of checking frequency for mode: The mode does not involve any calculation. It is found purely by observation — look for the value that repeats most.
What This Chapter Prepares You For
A solid understanding of mean, median, and mode is the foundation for everything that follows in this chapter. In the subsequent exercises, you will learn to organise raw data into frequency distribution tables, draw histograms, frequency polygons, and pie charts — all of which rely on the central tendency concepts introduced here.
These ideas carry forward into Class 9 and Class 10 as well. In Class 9, you study Statistics for grouped data, where you learn to calculate mean using the direct and assumed mean methods. In Class 10, Statistics becomes more advanced with cumulative frequency curves (ogives) and finding median graphically.
If you found this chapter's concepts on comparing quantities useful, you may also want to revisit Chapter 6 – Comparing Quantities Using Proportion, which deals with ratios and percentages — another common way of summarising data.