NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.3

1. Median \(= l + \dfrac{\frac n2 - cf}{f} \times h\), where the median class is the first whose cumulative frequency reaches \(\tfrac n2\).
2. \(\tfrac n2 = 34\); cumulative frequencies 4, 9, 22, 42: median class 125–145.
3. \[125 + \dfrac{34 - 22}{20} \times 20 = 137\]

1. Mean \(\bar x = \dfrac{\sum f_i x_i}{\sum f_i}\), with \(x_i\) the class mid-points.
2. \(\sum f_ix_i = 9320\): \(\bar x = \dfrac{9320}{68} \approx 137.05\).

1. Mode \[= l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\] for the modal class (highest frequency).
2. \[125 + \dfrac{20 - 13}{40 - 13 - 14} \times 20 = 125 + \dfrac{140}{13}\]
3. All three are close to 137 (mode a little lower).

1. \[\begin{aligned}&5 + x + 20 + 15 + y + 5 = 60 \\ \Rightarrow\ &x + y = 15\end{aligned}\]
2. 28.5 lies in 20–30, cf before it \(= 5 + x\): \[20 + \dfrac{30 - (5 + x)}{20} \times 10 = 28.5\]
3. \[\begin{aligned}&\dfrac{25 - x}{2} = 8.5 \\ \Rightarrow\ &x = 8\end{aligned}\], so \(y = 7\).

1. Convert to classes (15–20, 20–25, …, 55–60) with frequencies 2, 4, 18, 21, 33, 11, 3, 6, 2.
2. \(\tfrac n2 = 50\): median class 35–40 (cf before it 45, \(f = 33\)).
3. \[35 + \dfrac{50 - 45}{33} \times 5 \approx 35.76\]

1. Boundaries: 117.5–126.5, …, 171.5–180.5 (width 9).
2. \(\tfrac n2 = 20\); cf 3, 8, 17, 29: median class 144.5–153.5.
3. \[144.5 + \dfrac{20 - 17}{12} \times 9 = 146.75\]

Where marks slip: Forgetting to convert inclusive classes to boundaries is the most common lost mark.

1. Median \(= l + \dfrac{\frac n2 - cf}{f} \times h\), where the median class is the first whose cumulative frequency reaches \(\tfrac n2\).
2. \(\tfrac n2 = 200\); cf 14, 70, 130, 216: median class 3000–3500.
3. \[3000 + \dfrac{200 - 130}{86} \times 500 \approx 3406.98\]

1. Median \(= l + \dfrac{\frac n2 - cf}{f} \times h\), where the median class is the first whose cumulative frequency reaches \(\tfrac n2\).
2. \(\tfrac n2 = 50\); cf 6, 36, 76: median class 7–10.
3. \(7 + \dfrac{50 - 36}{40} \times 3 = 8.05\).

1. Mean \(\bar x = \dfrac{\sum f_i x_i}{\sum f_i}\), with \(x_i\) the class mid-points.
2. \(\sum f_ix_i = 832\): \(\bar x = 8.32\).

1. Mode \[= l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h\] for the modal class (highest frequency).
2. \[7 + \dfrac{40 - 30}{80 - 30 - 16} \times 3 = 7 + \dfrac{30}{34}\]

1. Median \(= l + \dfrac{\frac n2 - cf}{f} \times h\), where the median class is the first whose cumulative frequency reaches \(\tfrac n2\).
2. \(\tfrac n2 = 15\); cf 2, 5, 13, 19: median class 55–60.
3. \[55 + \dfrac{15 - 13}{6} \times 5 \approx 56.67\]