Learning Objective: Understand how the arithmetic mean acts as the balance center where the sum of distances on the left equals the sum of distances on the right.
Learning Objective: Explore how adding a fixed number to every data value shifts the mean by that number, and multiplying scales the mean.
Learning Objective: Compute weighted means accounting for data frequencies.
The arithmetic mean is not just a computational average; it represents the centre of data acting as a balance point. The sum of distances from the mean to all values lower than it (LHS) equals the sum of distances to all values higher than it (RHS).
When every value in a collection changes, the mean changes predictably:
When dealing with frequency data, the mean is calculated by multiplying each value by its frequency, summing those products, and dividing by the total sum of frequencies.
Spreadsheets are digital notebooks with rows and columns of cells used to organise data and compute statistics quickly using formulas like =SUM(range) and =AVERAGE(range).
Line graphs are exceptionally well-suited for illustrating trends and changes over time, connecting data points with line segments to show continuous variation (such as monthly temperatures, rainfall, and space launches).
1. Find the mean of the following data and share your observations:
- (i) First 50 natural numbers (1 to 50): Sum = (50 × 51) / 2 = 1275. Mean = 1275 / 50 = 25.5.
- (ii) First 50 odd numbers: Sum = 50 squared = 2500. Mean = 2500 / 50 = 50.
- (iii) First 50 multiples of 4: Sum = 4 × (50 × 51) / 2 = 5100. Mean = 5100 / 50 = 102.
2. Mark the missing dot so that the mean is 9 (Page 113):
Step 1: Calculate total sum required for all data points to have a mean of 9.
Step 2: Place the missing dot at the value that balances the LHS and RHS distance sums to zero.
3. Class average height is 150.2 cm, but students wore uniform shoes adding 1 cm:
- (i) Simpler way: Subtract 1 cm from the average rather than remeasuring every student.
- (ii) Correct average height = 150.2 - 1.0 = 149.2 cm (Option d).
5. Find the median of 8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92 (16 values):
Median is the average of the 8th and 9th sorted values (41 and 41) = 41.
7. Mean of 8, 13, 10, 4, 5, 20, y, 10 is 10.375. Find y:
Step 1: Sum = 8 + 13 + 10 + 4 + 5 + 20 + 10 + y = 70 + y.
Step 2: (70 + y) / 8 = 10.375 → 70 + y = 83 → y = 13.
8. Mean of 15 values is 134. Find the sum of the data:
Sum = Mean × Number of values = 134 × 15 = 2010.
9. Consider data: 12, 47, 8, 73, 18, 35, 39, 8, 29, 25, p. Median is 29. Which numbers could be p?
Sorted existing values (11 numbers): 8, 8, 12, 18, 25, 29, 35, 39, 47, 73. For 29 to be the median of 12 numbers, p must be placed such that 29 remains the middle value (e.g., any value between 25 and 35, or 29 itself).
Retail Price of Iodised Salt (Page 129):
- Students select 3 states and plot trends over 10 years using line graphs.
- Price comparisons show varying inflation and regional pricing patterns across states like Gujarat and Uttar Pradesh.
Household Energy Sources (Page 130):
- Analysis of HCES survey data shows rural households transitioning from kerosene to electricity from 1983 to 2023, while urban areas achieved near-universal electrification earlier.