Exam practice

$\overline{X} = \frac{810}{27} =$ 30 marks.

$\overline{X} = \frac{4550}{150} \approx$ 30.33 years.

$\overline{X} = \frac{2041}{35} \approx$ 58.31.

$\overline{X} = \frac{30000}{30} =$ 1000 ml (1 liter) a family.

cf: 5, 14, 29, 41, 47, 50, so N = 50.

Median $= \left(\frac{50 + 1}{2}\right)$th item = 25.5th item. Items 15 to 29 are 30, so the 25th and 26th are both 30: $M_d = 30$ marks.

cf: 6, 18, 36, 50, 57, 61, 62, so N = 62.

Median $= \left(\frac{62 + 1}{2}\right)$th item = 31.5th item. Items 19 to 36 are Rs 400, so the 31st and 32nd are both 400: $M_d =$ Rs 400.

cf: 5, 12, 22, 27, 30, so N = 30.

Median $= \left(\frac{30 + 1}{2}\right)$th item = 15.5th item. Items 13 to 22 are 30 kg: $M_d = 30$ kg.

cf: 10, 24, 44, 56, 60, so N = 60.

Median $= \left(\frac{60 + 1}{2}\right)$th item = 30.5th item. Items 25 to 44 are Rs 500: $M_d =$ Rs 500.

cf: 2, 7, 19, 26, 27, so N = 27. Median $= \left(\frac{27 + 1}{2}\right)$th item = 14th item. Items 8 to 19 are 30: $M_d = 30$ marks.

cf: 40, 70, 120, 140, 150, so N = 150. Median $= \left(\frac{150 + 1}{2}\right)$th item = 75.5th item. Items 71 to 120 are 35: $M_d = 35$ years.

cf: 4, 10, 20, 28, 33, 35, so N = 35. Median $= \left(\frac{35 + 1}{2}\right)$th item = 18th item. Items 11 to 20 are 55: $M_d = 55$.

cf: 8, 13, 22, 27, 30, so N = 30. Median $= \left(\frac{30 + 1}{2}\right)$th item = 15.5th item. Items 14 to 22 are 1000 ml: $M_d = 1000$ ml.

fX: 50, 180, 450, 480, 300, 180; $\sum fX = 1640$, N = 50.

$\overline{X} = \frac{1640}{50} =$ 32.8 marks.

fX: 1800, 4200, 7200, 6300, 3500, 2200, 600; $\sum fX = 25800$, N = 62.

$\overline{X} = \frac{25800}{62} \approx$ Rs 416.13.

fX: 100, 175, 300, 175, 120; $\sum fX = 870$, N = 30.

$\overline{X} = \frac{870}{30} =$ 29 kg.

fX: 3000, 5600, 10000, 7200, 2800; $\sum fX = 28600$, N = 60.

$\overline{X} = \frac{28600}{60} \approx$ Rs 476.67.

N = 21 + k, $\sum fX = 60 + 150 + 35k + 450 + 165 = 825 + 35k$.

$$\begin{aligned} 38(21 + k) &= 825 + 35k \\ 798 + 38k &= 825 + 35k \\ 3k &= 27 \end{aligned}$$

k = 9. Check ✓: $\frac{825 + 315}{30} = \frac{1140}{30} = 38$.

N = 14 + m, $\sum fX = 465 + 165m + 875 + 740 + 390 = 2470 + 165m$.

$$\begin{aligned} 175(14 + m) &= 2470 + 165m \\ 2450 + 175m &= 2470 + 165m \\ 10m &= 20 \end{aligned}$$

m = 2. Check ✓: $\frac{2470 + 330}{16} = \frac{2800}{16} = 175$.

N = 20, $\sum fX = 80 + 88 + 125 + 104 + 2r + 34 = 431 + 2r$.

$$\begin{aligned} \frac{431 + 2r}{20} &= 24.55 \\ 431 + 2r &= 491 \\ 2r &= 60 \end{aligned}$$

r = 30 kg. This time the unknown is a value, not a frequency, but the method is the same.

The highest frequency is 6, for 20: $M_o = 20$.

The highest frequency is 17, for Rs 175: $M_o =$ Rs 175.

The highest frequency is 12, for 45: $M_o = 45$ marks.

The highest frequency is 15, for 30 kg: $M_o = 30$ kg.

Mean: $\overline{X} = \frac{1160}{40} =$ 29 cm.

Median: the $\left(\frac{40 + 1}{2}\right)$th item = 20.5th item. Items 17 to 28 are 30, so the 20th and 21st are both 30: $M_d = 30$ cm.

(The mode is 30 cm too, with 12 plants. The mean is a little lower, 29 cm, because the one 10 cm seedling pulls it down.)

In order: 15, 20, 30, 40, 60, 80, 90; N = 7.
Q₁: the $\frac{8}{4} =$ 2nd item, 20. Q₃: the $\frac{3 \times 8}{4} =$ 6th item, 80.

In order: 200, 275, 320, 350, 370, 400, 550, 600, 625, 650, 700; N = 11.
Q₁: the $\frac{12}{4} =$ 3rd item, Rs 320. Q₃: the $\frac{3 \times 12}{4} =$ 9th item, Rs 625.

In order: 5, 10, 15, 20, 25; N = 5.
Q₁: the $\frac{6}{4} =$ 1.5th item = 5 + 0.5 × (10 − 5) = 7.5.
Q₃: the $\frac{18}{4} =$ 4.5th item = 20 + 0.5 × (25 − 20) = 22.5.

In order: 9, 12, 15, 18, 21, 24; N = 6.
Q₁: the $\frac{7}{4} =$ 1.75th item = 9 + 0.75 × (12 − 9) = 11.25 kg.
Q₃: the $\frac{21}{4} =$ 5.25th item = 21 + 0.25 × (24 − 21) = 21.75 kg.

In order: 12, 16, 18, 21, 23, 25, 26, 28, 29; N = 9.
Q₁: the $\frac{10}{4} =$ 2.5th item = 16 + 0.5 × (18 − 16) = 17 ∘°C.
Q₃: the $\frac{30}{4} =$ 7.5th item = 26 + 0.5 × (28 − 26) = 27 ∘°C.

cf: 3, 8, 16, 23, 27, 30; N = 30.
Q₁: the $\frac{31}{4} =$ 7.75th item. The 7th and 8th are both 10: Q₁ = 10.
Q₃: the $\frac{93}{4} =$ 23.25th item. The 23rd is 20, the 24th is 25: 20 + 0.25 × (25 − 20) = Q₃ = 21.25.

The wages are not in order. In ascending order with their frequencies:

Q₁: the $\frac{29}{4} =$ 7.25th item. The 7th and 8th are both 300: Q₁ = Rs 300.
Q₃: the $\frac{87}{4} =$ 21.75th item. The 21st and 22nd are both 450: Q₃ = Rs 450.

cf: 8, 18, 25, 37, 44, 50; N = 50.
Q₁: the $\frac{51}{4} =$ 12.75th item. The 12th and 13th are both 115: Q₁ = 115 kg.
Q₃: the $\frac{153}{4} =$ 38.25th item. The 38th and 39th are both 145: Q₃ = 145 kg.

In ascending order:

Q₁: the $\frac{29}{4} =$ 7.25th item. The 7th and 8th are both 14: Q₁ = 14.
Q₃: the $\frac{87}{4} =$ 21.75th item. The 21st and 22nd are both 18: Q₃ = 18.

Q₁: the $\frac{20}{4} =$ 5th item, in 15 (items 4 to 7): Q₁ = 15.
Q₃: the $\frac{3 \times 20}{4} =$ 15th item, in 25 (items 14 to 17): Q₃ = 25.

Lower quartile Q₁: the $\frac{46}{4} =$ 11.5th item. The 11th and 12th are both 27: Q₁ = 27 kg.
Upper quartile Q₃: the $\frac{3 \times 46}{4} =$ 34.5th item. The 34th is 35 and the 35th is 38: 35 + 0.5 × (38 − 35) = Q₃ = 36.5 kg.

In order: 47, 49, 51, 52, 53, 54, 55; N = 7.
Q₁ is the 2nd item, 49; Q₃ the 6th, 54.

In order: 4, 5, 6, 6, 7, 7, 9, 9, 10; N = 9.
Q₁: the $\frac{10}{4} =$ 2.5th item = 5 + 0.5 × (6 − 5) = 5.5.
Q₃: the $\frac{30}{4} =$ 7.5th item; the 7th and 8th are both 9: 9.

N = 5. Q₁: the 1.5th item = 5 + 0.5 × 5 = 7.5. Q₃: the 4.5th item = 20 + 0.5 × 5 = 22.5. (The same data as 1c, in order.)

In order: 56, 60, 64, 68, 70, 72, 80, 85; N = 8.
Q₁: the $\frac{9}{4} =$ 2.25th item = 60 + 0.25 × (64 − 60) = 61.
Q₃: the $\frac{27}{4} =$ 6.75th item = 72 + 0.75 × (80 − 72) = 78.

cf: 5, 12, 24, 39, 57, 60, 62; N = 62.
Q₁: the $\frac{63}{4} =$ 15.75th item; the 15th and 16th are both 80: Q₁ = 80.
Q₃: the $\frac{189}{4} =$ 47.25th item; the 47th and 48th are both 90: Q₃ = 90.

cf: 40, 60, 75, 92, 117, 127, 132; N = 132.
Q₁: the $\frac{133}{4} =$ 33.25th item; the 33rd and 34th are both 300: Q₁ = Rs 300.
Q₃: the $\frac{399}{4} =$ 99.75th item; the 99th and 100th are both 700: Q₃ = Rs 700.

cf: 5, 15, 27, 44, 60, 63, 67; N = 67.
Q₁: the $\frac{68}{4} =$ 17th item, in 122 (items 16 to 27): Q₁ = 122 cm.
Q₃: the $\frac{3 \times 68}{4} =$ 51st item, in 124 (items 45 to 60): Q₃ = 124 cm.

cf: 4, 10, 18, 23, 26, 30; N = 30.
Q₁: the $\frac{31}{4} =$ 7.75th item; the 7th and 8th are both 22: Q₁ = 22 ∘°C.
Q₃: the $\frac{93}{4} =$ 23.25th item; the 23rd is 26 and the 24th is 27: 26 + 0.25 × (27 − 26) = Q₃ = 26.25 ∘°C.