Chapter 10 — How Quantities Combine: Understanding Data

Chapter 10, 'How Quantities Combine: Understanding Data', covers weighted averages — how to correctly combine data sets of different sizes — and data visualisation using stacked bar charts and 100% stacked bar charts.

The weighted mean of values x₁, x₂, …, xn with weights w₁, w₂, …, wn is (w₁x₁ + w₂x₂ + … + wnxn) divided by (w₁ + w₂ + … + wn). Each value is multiplied by how large or important it is before summing.

Simply averaging two group averages treats both groups as equally large, which is wrong when they differ in size. The correct method weights each group's average by its size — illustrated in the chapter with a badminton academy of 8 seniors and 3 juniors.

When all groups (collections) have the same number of values, each average carries equal weight, so the simple arithmetic mean of those averages gives the correct combined average. The chapter demonstrates this with Jaspreet's cycling data, where she cycled exactly 5 days each week for 3 weeks.

The concentration of a mixture equals the weighted mean of the individual concentrations, with the volume or mass of each component as its weight. For example, mixing 500 mL at 10% jaggery with 200 mL at 20% gives a concentration of (500×0.1 + 200×0.2) / (500+200) ≈ 13%.

The chapter states that Brahmagupta recorded the formula in his work Brāhmasphuṭasiddhānta (c. 628 CE), where he used it to define the mean depth of an irregularly shaped pit. Śrīdharācārya (c. 750 CE) also used it to estimate the purity of gold alloys.

A stacked bar chart joins individual bars end-to-end within each cluster so the full bar length shows the total while the component bars show each part's contribution. The chapter recommends it when comparing totals across groups is as important as tracking the components that make up those totals.

A 100% stacked bar chart rescales every stacked bar to a fixed length, with each smaller bar's length proportional to its share of the total — making it similar to a pie chart. The chapter notes that comparing lengths in a bar chart may be easier than comparing angles in pie charts, though both show proportions rather than absolute values.

You cannot compare actual quantities across different bars, only the proportions within each bar. The chapter shows that if Fatima's garden has a higher summer percentage than Naveen's, this does not mean she had more blooms in summer — Naveen's larger total could produce more actual flowers even from a smaller percentage.

No. The chapter states that given a stacked bar chart you can make a corresponding 100% stacked bar chart, but the reverse is not possible because the 100% form discards the actual totals.

When exam components have different importance — for example internals, project, and final exam in the ratio 3 : 2 : 5 — the annual score is the weighted mean of the component percentages using those ratios as weights. In the chapter's example, Rehmat's weighted annual score is 67.3%, compared to a simple average of 65.6%.

Yes — the full NCERT textbook PDF for Class 9 Maths (Ganita Manjari) is freely available on cbseprepmaster.com with no account or sign-up required.