How Quantities Combine: Understanding Data Worksheet

(a) 2

1. The mode is the value with the greatest frequency.
2. 2 occurs twice; every other listed value occurs once.

(c) 15

1. The weights are equal.
2. The weighted mean reduces to (12 + 18)/2 = 15.

(b) 100% of its own group

1. Each group’s category frequencies are converted to percentages of that group.
2. The complete bar is 100%, even when the group sizes differ.

(b) 70

1. Total marks = 10 × 50 + 20 × 80 = 2100.
2. Divide by 30 students: 2100/30 = 70.

Combined average 72.

1. The total scores are 20 × 60 = 1200 and 30 × 80 = 2400.
2. Combined average = (1200 + 2400)/(20 + 30) = 3600/50 = 72.
3. The classes have different sizes; weight each average by its student count instead of treating the averages equally.

₹48/kg.

1. Total cost is 3 × 40 + 2 × 60 = ₹240.
2. Total mass is 3 + 2 = 5 kg.
3. The weighted mean cost is 240/5 = ₹48/kg, not the unweighted average ₹50/kg.

Mean 7, median 6, mode 6.

1. The sum is 35; dividing by 5 gives mean 7.
2. The third value in this sorted list is 6, so median 6.
3. 6 occurs most often, giving mode 6. The measures use different features of the data.

Mean 20, median 15, mode 15 minutes; the median is less affected by the exceptional delay.

1. The total is 100 minutes for 5 trips, so the mean is 20.
2. The middle sorted value is 15, giving the median; 15 also occurs most often, giving the mode.
3. The unusually long trip raises the mean. The median gives a useful typical value here, though the right measure depends on the purpose.
4. An investigative question could be “How does travel time change by day or route?” Record comparable trips and organise them in a table.

Ordinary bar totals 20 and 30; percentage bars show A 60%/40% and B 50%/50%.

1. An ordinary stacked bar combines category counts: A = 12 + 8 = 20, B = 15 + 15 = 30.
2. Divide each category by its own class total to form percentage shares.
3. For A, bus share 12/20 = 60%, walking 40%; for B the shares are 50% each.
4. 100% stacked bars have equal total height and directly compare composition; ordinary bars retain absolute totals.

Mean 20 min; median 12.5 min; mode 10 min.

1. The sum is 120, so the mean is 120/6 = 20.
2. There are six sorted values; average the third and fourth: (10 + 15)/2 = 12.5.
3. 10 occurs most frequently, so it is the mode.
4. A small or selected group may not represent all ages/classes. Use a fair sampling plan, a clear question and a consistent measurement period.

X total 50, Y total 60; both have proportions 60% bus, 20% bicycle, 20% walking.

1. Ordinary stacked bars show different totals: 50 and 60.
2. For X divide each count by 50; for Y divide by 60.
3. Both percentage bars have segments 60%, 20% and 20%, hence identical composition.
4. Equal composition does not imply equal absolute numbers. Nor does this observational table alone establish the cause of transport choices.

Totals 30 and 40; both percentage bars are 40% cycling and 60% walking.

1. Ordinary stacked bars have heights 30 and 40, divided into the stated counts.
2. A cycling share is 12/30 = 40%; B cycling share is 16/40 = 40%.
3. Both walking shares are 60%; percentage bars therefore look identical.
4. Equal proportions can belong to different total sizes. Ordinary bars preserve the distinction in counts.