Averages and Measures of Spread (Copy)
1.
Find the mean of the data: 5, 7, 9, 11, 13.
2.
Find the median of the data in Q1.
3.
Find the mode of the data: 2, 3, 5, 3, 6, 3, 7.
4.
Find the range of the data: 4, 8, 10, 15, 20.
5.
The marks of 6 students are: 10, 12, 14, 16, 18, 20. Find the mean.
6.
From Q5, find the median.
7.
From Q5, find the range.
8.
The ages of 7 people are: 21, 25, 29, 33, 25, 30, 25. Find the mode.
9.
A student’s weekly pocket money over 5 weeks is: $5, $7, $8, $10, $5. Find the mean.
10.
From Q9, find the median.
11.
From Q9, find the mode.
12.
The daily sales (in items) of a shop are: 50, 60, 70, 80, 90. Find the range.
13.
The scores of 8 students are: 15, 18, 20, 22, 25, 20, 18, 15. Find the mean.
14.
From Q13, find the median.
15.
From Q13, find the mode.
16.
From Q13, find the range.
17.
The grouped data shows heights (cm) of 40 students:
140–149: 5
150–159: 12
160–169: 15
170–179: 8
Estimate the mean height.
18.
From Q17, identify the modal class.
19.
From Q17, find the class interval containing the median.
20.
The grouped data shows test scores:
0–9: 4
10–19: 6
20–29: 10
30–39: 8
40–49: 2
Estimate the mean score.
21.
From Q20, identify the modal class.
22.
From Q20, find the class containing the median.
23.
The grouped data shows weights (kg):
40–49: 5
50–59: 7
60–69: 13
70–79: 10
80–89: 5
Estimate the mean weight.
24.
From Q23, find the modal class.
25.
From Q23, find the class containing the median.
26.
The daily temperatures (°C) for a week are: 28, 30, 32, 31, 29, 33, 30. Find the mean.
27.
From Q26, find the median.
28.
From Q26, find the mode.
29.
From Q26, find the range.
30.
Explain why the median may be more useful than the mean when interpreting skewed data (e.g. income).
1.
Mean = (5+7+9+11+13)/5 = 45/5 = 9.
2.
Ordered: 5,7,9,11,13 → Middle = 9.
3.
Mode = most frequent = 3 (appears 3 times).
4.
Range = 20−4 = 16.
5.
Mean = (10+12+14+16+18+20)/6 = 90/6 = 15.
6.
Ordered: 10,12,14,16,18,20 → Median = (14+16)/2 = 15.
7.
Range = 20−10 = 10.
8.
Mode = most frequent value = 25 (appears 3 times).
9.
Mean = (5+7+8+10+5)/5 = 35/5 = 7.
10.
Ordered: 5,5,7,8,10 → Median = 7.
11.
Mode = 5 (appears twice).
12.
Range = 90−50 = 40.
13.
Sum = 15+18+20+22+25+20+18+15 = 153.
Mean = 153/8 = 19.125.
14.
Ordered: 15,15,18,18,20,20,22,25 → Median = (18+20)/2 = 19.
15.
Modes = 15,18,20 (all appear twice). → Multimodal.
16.
Range = 25−15 = 10.
17.
Use midpoints: 144.5, 154.5, 164.5, 174.5.
Mean ≈ (5×145 + 12×155 + 15×165 + 8×175)/40.
= (725+1860+2475+1400)/40=6460/40=161.5 cm.
18.
Modal class = highest frequency = 160–169.
19.
Median position = 40/2=20th.
Cumulative freq: up to 149=5, up to 159=17, up to 169=32.
So median lies in 160–169.
20.
Midpoints: 4.5,14.5,24.5,34.5,44.5.
Mean=(4×4.5+6×14.5+10×24.5+8×34.5+2×44.5)/30.
= (18+87+245+276+89)/30=715/30=23.8.
21.
Modal class = 20–29 (highest frequency=10).
22.
Median position = 30/2=15th.
Cumulative: up to 9=4, up to 19=10, up to 29=20 → 15th lies in 20–29.
23.
Midpoints: 44.5,54.5,64.5,74.5,84.5.
Mean=(5×44.5+7×54.5+13×64.5+10×74.5+5×84.5)/40.
= (222.5+381.5+838.5+745+422.5)/40=2610/40=65.25 kg.
24.
Modal class = 60–69 (highest frequency=13).
25.
Median position = 40/2=20th.
Cumulative: 5,12,25,35,40 → 20th in 60–69.
26.
Mean = (28+30+32+31+29+33+30)/7 = 213/7= 30.4°C.
27.
Ordered: 28,29,30,30,31,32,33. Median = 30 (middle).
28.
Mode = 30 (appears twice).
29.
Range = 33−28= 5.
30.
Median is resistant to extreme values.
Example: incomes 1000,1200,1500,50000 → mean=13425 (distorted by 50000), median=1350 (better central value).