Class 9 Maths Chapter 10: How Quantities Combine: Understanding Data — Notes, Examples & Practice

The mean (average) of a set of values is their total divided by how many there are. To combine two groups, go back to totals: total of all values ÷ total number of values.

If class A (30 students) averages 60 marks and class B (20 students) averages 75, the combined mean is not $\dfrac{60 + 75}{2} = 67.5$. Totals: $30 \times 60 = 1800$ and $20 \times 75 = 1500$, so the combined mean is $\dfrac{3300}{50} = 66$. The bigger class pulls the average towards its own.

The simple average of two averages is right only when the groups are the same size.

In a weighted mean, each value counts according to its weight — how important it is or how often it occurs. Multiply each value by its weight, add, then divide by the sum of the weights.

Schools combine marks this way: if the final exam counts 60%, projects 25% and class tests 15%, a student with 70, 90 and 80 gets $\dfrac{0.6(70) + 0.25(90) + 0.15(80)}{1} = 42 + 22.5 + 12 = 76.5$. Put all components on a common scale (e.g. out of 100) first.

Two useful facts: multiplying all weights by the same number does not change the mean; making all weights equal gives the ordinary mean. A cricket team's run rate (total runs ÷ total overs) is also a weighted mean of the over-by-over rates.

The concentration of a mixture is (amount of ingredient) ÷ (total amount). When you mix two solutions, find the amount of ingredient in each, add them, and divide by the total volume — again, totals, not averages of percentages.

Alloys work the same way. Gold purity is measured in karats out of 24 (24K is pure gold; 18K is $\dfrac{18}{24} = 75\%$ gold). Traditional Indian texts used a similar purity scale called varṇa.

Mixing 1 litre of 10% sugar syrup with 3 litres of 30% syrup does not give 20%. Sugar: $0.1 + 0.9 = 1$ litre in 4 litres → 25%.

Adding a new value above the mean raises it; adding one below the mean lowers it; adding a value equal to the mean leaves it unchanged. Removing values works the other way round.

You can predict the size of the change: if $n$ values have mean $m$ and you add a value $x$, the new mean is $m + \dfrac{x - m}{n + 1}$. The new value's 'extra' $(x - m)$ is shared among all $n + 1$ values.

Adding the same constant to every value raises the mean by that constant; multiplying every value by $k$ multiplies the mean by $k$. A family's average age, for instance, rises by exactly 1 every year if nobody joins or leaves.

A clustered bar chart puts bars side by side to compare categories. A stacked bar chart divides each bar into coloured parts, so you can see both the total (whole bar height) and the parts (segment heights).

A 100% stacked bar chart makes every bar the same height (100%), each split into percentages. It shows proportions, not amounts — it hides the totals.

So be careful: in a 100% chart, school A's 40% girls might be more students than school B's 60% girls if A is much larger. Before you conclude, ask: does the chart show amounts or shares? What extra information would I need?