Chapter 13: Statistics

Solution

The modal class has the highest frequency.

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{47+42+36+37+20}{5} = 36.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{49+20+20+40+39}{5} = 33.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{18+46+28+38+10}{5} = 28.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{23+29+25+47+16}{5} = 28.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{50+18+22+13+35}{5} = 27.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{28+26+46+25+20}{5} = 29.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{16+49+21+38+18}{5} = 28.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{22+33+35+50+34}{5} = 34.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{42+27+47+50+22}{5} = 37.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{33+37+12+23+47}{5} = 30.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{29+22+42+30+20}{5} = 28.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{35+12+18+40+46}{5} = 30.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{29+34+49+28+23}{5} = 32.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{33+20+28+27+27}{5} = 27.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{20+28+29+38+10}{5} = 25.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{45+42+17+46+13}{5} = 32.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{30+45+20+11+36}{5} = 28.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{17+44+11+50+12}{5} = 26.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{27+28+43+11+17}{5} = 25.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{27+22+39+44+16}{5} = 29.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{40+20+38+19+30}{5} = 29.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{25+13+37+16+22}{5} = 22.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{23+16+34+45+44}{5} = 32.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{25+13+50+39+42}{5} = 33.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{43+46+18+12+42}{5} = 32.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{43+36+38+46+30}{5} = 38.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{12+30+13+29+11}{5} = 19.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{30+49+17+39+24}{5} = 31.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{29+24+23+12+15}{5} = 20.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{23+11+22+41+14}{5} = 22.2\)

Solution

Mode is the most frequently occurring value.

40 appears 4 times (most frequent), so mode = 40

Solution

Mode is the most frequently occurring value.

43 appears 4 times (most frequent), so mode = 43

Solution

Mode is the most frequently occurring value.

32 appears 4 times (most frequent), so mode = 32

Solution

Mode is the most frequently occurring value.

37 appears 4 times (most frequent), so mode = 37

Solution

Mode is the most frequently occurring value.

47 appears 4 times (most frequent), so mode = 47

Solution

Mode is the most frequently occurring value.

47 appears 4 times (most frequent), so mode = 47

Solution

Mode is the most frequently occurring value.

50 appears 4 times (most frequent), so mode = 50

Solution

Mode is the most frequently occurring value.

44 appears 4 times (most frequent), so mode = 44

Solution

Mode is the most frequently occurring value.

40 appears 4 times (most frequent), so mode = 40

Solution

Mode is the most frequently occurring value.

47 appears 4 times (most frequent), so mode = 47

Solution

Mode is the most frequently occurring value.

34 appears 4 times (most frequent), so mode = 34

Solution

Mode is the most frequently occurring value.

47 appears 4 times (most frequent), so mode = 47

Solution

Mode is the most frequently occurring value.

50 appears 4 times (most frequent), so mode = 50

Solution

Mode is the most frequently occurring value.

46 appears 4 times (most frequent), so mode = 46

Solution

Mode is the most frequently occurring value.

50 appears 4 times (most frequent), so mode = 50

Solution

Mode is the most frequently occurring value.

38 appears 4 times (most frequent), so mode = 38

Solution

Mode is the most frequently occurring value.

39 appears 4 times (most frequent), so mode = 39

Solution

Mode is the most frequently occurring value.

36 appears 4 times (most frequent), so mode = 36

Solution

Mode is the most frequently occurring value.

44 appears 4 times (most frequent), so mode = 44

Solution

Mode is the most frequently occurring value.

37 appears 4 times (most frequent), so mode = 37

Solution

Mode is the most frequently occurring value.

35 appears 4 times (most frequent), so mode = 35

Solution

Mode is the most frequently occurring value.

41 appears 4 times (most frequent), so mode = 41

Solution

Mode is the most frequently occurring value.

46 appears 4 times (most frequent), so mode = 46

Solution

Mode is the most frequently occurring value.

45 appears 4 times (most frequent), so mode = 45

Solution

Mode is the most frequently occurring value.

31 appears 4 times (most frequent), so mode = 31

Solution

Mode is the most frequently occurring value.

36 appears 4 times (most frequent), so mode = 36

Solution

Mode is the most frequently occurring value.

44 appears 4 times (most frequent), so mode = 44

Solution

Mode is the most frequently occurring value.

31 appears 4 times (most frequent), so mode = 31

Solution

Mode is the most frequently occurring value.

48 appears 4 times (most frequent), so mode = 48

Solution

Mode is the most frequently occurring value.

46 appears 4 times (most frequent), so mode = 46

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{50+29+43+28+49}{5} = 39.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{16+23+45+36+27}{5} = 29.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{43+19+45+48+37}{5} = 38.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{34+22+24+47+34}{5} = 32.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{38+28+42+24+39}{5} = 34.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{20+10+35+29+20}{5} = 22.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{42+47+35+19+49}{5} = 38.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{46+29+22+22+37}{5} = 31.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{32+41+16+46+44}{5} = 35.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{32+46+20+43+30}{5} = 34.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{21+36+38+18+47}{5} = 32.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{10+35+42+34+47}{5} = 33.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{22+50+26+39+33}{5} = 34.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{27+50+34+39+39}{5} = 37.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{36+41+12+37+25}{5} = 30.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{29+23+28+41+41}{5} = 32.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{11+36+13+13+46}{5} = 23.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{31+21+36+33+36}{5} = 31.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{25+34+30+23+36}{5} = 29.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{11+19+34+13+30}{5} = 21.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{24+19+11+47+34}{5} = 27.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{22+25+20+45+26}{5} = 27.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{31+19+36+18+48}{5} = 30.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{32+13+16+33+25}{5} = 23.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{28+44+30+43+47}{5} = 38.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{17+50+15+16+27}{5} = 25.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{32+28+35+26+28}{5} = 29.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{24+49+10+13+37}{5} = 26.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{23+38+45+47+31}{5} = 36.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{41+14+41+27+46}{5} = 33.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{46+12+15+49+20}{5} = 28.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{33+29+16+16+26}{5} = 24.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{26+46+35+28+11}{5} = 29.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{17+25+41+19+39}{5} = 28.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{41+19+31+47+40}{5} = 35.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{34+29+21+46+48}{5} = 35.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{46+22+32+29+42}{5} = 34.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{15+42+11+37+35}{5} = 28.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{41+49+10+44+37}{5} = 36.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{25+18+46+26+12}{5} = 25.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{37+19+42+22+11}{5} = 26.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{21+14+47+35+27}{5} = 28.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{34+44+26+33+50}{5} = 37.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{42+46+41+13+41}{5} = 36.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{11+42+24+30+28}{5} = 27.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{48+20+25+38+25}{5} = 31.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{49+18+33+14+37}{5} = 30.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{33+50+45+31+11}{5} = 34.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{29+34+43+43+20}{5} = 33.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{13+27+12+48+12}{5} = 22.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{49+15+13+14+43}{5} = 26.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{12+29+19+31+44}{5} = 27.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{39+14+31+30+40}{5} = 30.8\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{25+20+15+25+28}{5} = 22.6\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{12+21+11+23+18}{5} = 17.0\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{10+39+14+21+47}{5} = 26.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{24+10+42+26+19}{5} = 24.2\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{27+46+48+21+30}{5} = 34.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{38+41+47+11+40}{5} = 35.4\)

Solution

Mean \(= \frac{\text{Sum of observations}}{\text{Number of observations}}\)

\(= \frac{36+30+43+34+19}{5} = 32.4\)

Solution

Mode is the most frequently occurring value.

42 appears 4 times (most frequent), so mode = 42

Solution

Mode is the most frequently occurring value.

39 appears 4 times (most frequent), so mode = 39

Solution

Mode is the most frequently occurring value.

37 appears 4 times (most frequent), so mode = 37

Solution

Mode is the most frequently occurring value.

35 appears 4 times (most frequent), so mode = 35

Solution

Mode is the most frequently occurring value.

44 appears 4 times (most frequent), so mode = 44

Solution

Mode is the most frequently occurring value.

49 appears 4 times (most frequent), so mode = 49

Solution

Mode is the most frequently occurring value.

39 appears 4 times (most frequent), so mode = 39

Solution

Mode is the most frequently occurring value.

31 appears 4 times (most frequent), so mode = 31

Solution

Mode is the most frequently occurring value.

32 appears 4 times (most frequent), so mode = 32

Solution

Mode is the most frequently occurring value.

39 appears 4 times (most frequent), so mode = 39

Solution

Mode is the most frequently occurring value.

38 appears 4 times (most frequent), so mode = 38

Solution

Mode is the most frequently occurring value.

40 appears 4 times (most frequent), so mode = 40

Solution

Mode is the most frequently occurring value.

41 appears 4 times (most frequent), so mode = 41

Solution

Mode is the most frequently occurring value.

38 appears 4 times (most frequent), so mode = 38

Solution

Mode is the most frequently occurring value.

43 appears 4 times (most frequent), so mode = 43

Solution

Mode is the most frequently occurring value.

42 appears 4 times (most frequent), so mode = 42

Solution

Mode is the most frequently occurring value.

37 appears 4 times (most frequent), so mode = 37

Solution

Mode is the most frequently occurring value.

48 appears 4 times (most frequent), so mode = 48

Solution

Mode is the most frequently occurring value.

46 appears 4 times (most frequent), so mode = 46

Solution

Mode is the most frequently occurring value.

35 appears 4 times (most frequent), so mode = 35

Solution

Mode is the most frequently occurring value.

41 appears 4 times (most frequent), so mode = 41

Solution

Mode is the most frequently occurring value.

33 appears 4 times (most frequent), so mode = 33

Solution

Mode is the most frequently occurring value.

50 appears 4 times (most frequent), so mode = 50

Solution

Mode is the most frequently occurring value.

35 appears 4 times (most frequent), so mode = 35

Solution

Mode is the most frequently occurring value.

34 appears 4 times (most frequent), so mode = 34

Solution

Mode is the most frequently occurring value.

43 appears 4 times (most frequent), so mode = 43

Solution

Mode is the most frequently occurring value.

38 appears 4 times (most frequent), so mode = 38

Solution

Mode is the most frequently occurring value.

39 appears 4 times (most frequent), so mode = 39

Solution

Mode is the most frequently occurring value.

37 appears 4 times (most frequent), so mode = 37

Solution

Mode is the most frequently occurring value.

41 appears 4 times (most frequent), so mode = 41

Solution

Mode is the most frequently occurring value.

49 appears 4 times (most frequent), so mode = 49

Solution

Mode is the most frequently occurring value.

41 appears 4 times (most frequent), so mode = 41

Solution

Mode is the most frequently occurring value.

43 appears 4 times (most frequent), so mode = 43

Solution

Mode is the most frequently occurring value.

50 appears 4 times (most frequent), so mode = 50

Solution

Mode is the most frequently occurring value.

50 appears 4 times (most frequent), so mode = 50

Solution

Mode is the most frequently occurring value.

35 appears 4 times (most frequent), so mode = 35

Solution

Mode is the most frequently occurring value.

31 appears 4 times (most frequent), so mode = 31

Solution

Mode is the most frequently occurring value.

37 appears 4 times (most frequent), so mode = 37

Solution

Mode is the most frequently occurring value.

32 appears 4 times (most frequent), so mode = 32

Solution

Mode is the most frequently occurring value.

46 appears 4 times (most frequent), so mode = 46

Solution

Mode is the most frequently occurring value.

48 appears 4 times (most frequent), so mode = 48

Solution

Mode is the most frequently occurring value.

32 appears 4 times (most frequent), so mode = 32

Solution

Mode is the most frequently occurring value.

47 appears 4 times (most frequent), so mode = 47

Solution

Mode is the most frequently occurring value.

32 appears 4 times (most frequent), so mode = 32

Solution

Mode is the most frequently occurring value.

32 appears 4 times (most frequent), so mode = 32

Solution

Mode is the most frequently occurring value.

46 appears 4 times (most frequent), so mode = 46

Solution

Mode is the most frequently occurring value.

45 appears 4 times (most frequent), so mode = 45

Solution

Mode is the most frequently occurring value.

34 appears 4 times (most frequent), so mode = 34

Solution

Mode is the most frequently occurring value.

43 appears 4 times (most frequent), so mode = 43

Solution

Mode is the most frequently occurring value.

44 appears 4 times (most frequent), so mode = 44

Solution

Mode is the most frequently occurring value.

34 appears 4 times (most frequent), so mode = 34

Solution

Mode is the most frequently occurring value.

41 appears 4 times (most frequent), so mode = 41

Solution

Mode is the most frequently occurring value.

36 appears 4 times (most frequent), so mode = 36

Solution

Mode is the most frequently occurring value.

44 appears 4 times (most frequent), so mode = 44

Solution

Mode is the most frequently occurring value.

43 appears 4 times (most frequent), so mode = 43

Solution

Mode is the most frequently occurring value.

33 appears 4 times (most frequent), so mode = 33

Solution

Mode is the most frequently occurring value.

49 appears 4 times (most frequent), so mode = 49

Solution

Mode is the most frequently occurring value.

47 appears 4 times (most frequent), so mode = 47

Solution

Mode is the most frequently occurring value.

37 appears 4 times (most frequent), so mode = 37

Solution

Mode is the most frequently occurring value.

43 appears 4 times (most frequent), so mode = 43

Solution

Class marks: 17.5, 32.5, 47.5, 62.5, 77.5.

Σfᵢxᵢ = 35+97.5+332.5+375+465 = 1305. Σfᵢ = 24.

Mean = 1305/24 = 54.375.

Let me recheck: 2×17.5=35, 3×32.5=97.5, 7×47.5=332.5, 6×62.5=375, 6×77.5=465. Sum=1305. N=24. Mean=1305/24=54.375.

Solution

Modal class: 20-30 (f=15). l=20, f₁=15, f₀=8, f₂=12, h=10.

Mode = 20 + (15−8)/(30−8−12) × 10 = 20 + 7/10 × 10 = 20+7 = 27.

Wait: 2f₁−f₀−f₂ = 30−8−12 = 10. Mode = 20 + 70/10 = 27.

Solution

The median class is identified as the class whose cumulative frequency is ≥ n/2 for the first time.

Solution

The intersection of the two ogives gives the median.

Solution

Sum of 5 = 150. Sum of 4 = 112. Excluded = 150−112 = 38.

Solution

An ogive plots cumulative frequency against upper (or lower) class boundaries.

Solution

dᵢ = xᵢ − a, where a is the assumed mean and xᵢ is the class mark.

Solution

N = 46, n/2 = 23. CF: 5, 13, 28, 40, 46. Median class: 20-30 (CF first ≥ 23).

l = 20, cf = 13, f = 15, h = 10. Median = 20 + (23−13)/15 × 10 = 20 + 100/15 = 20 + 6.67 = 26.67.

Solution

Marks: 10, 30, 50, 70, 90. Σfᵢxᵢ = 170+30f+1600+70f+1710 = 3480+100f.

Σfᵢ = 68+2f. Mean=50: (3480+100f)/(68+2f) = 50.

3480+100f = 3400+100f → 3480=3400. This gives 80=0, which is impossible. Let me recheck.

3480+100f = 50(68+2f) = 3400+100f → 3480 = 3400. Error in data. Fixing: If frequencies are 17, f₁, 32, f₂, 19 with f₁=f₂=f:

Actually the computation is correct and shows an inconsistency. Let's try different class marks or re-examine. With correct NCERT-style problem: Σfx = 170+30f+1600+70f+1710 = 3480+100f. N = 68+2f. 50 = (3480+100f)/(68+2f).

50(68+2f) = 3480+100f → 3400+100f = 3480+100f → 0 = 80. The equation has no solution with equal f values. This means f₁ ≠ f₂. Taking them as different: 17+f₁+32+f₂+19 = 120 → f₁+f₂ = 52.

Mean: (170+30f₁+1600+70f₂+1710)/120 = 50 → 3480+30f₁+70f₂ = 6000 → 30f₁+70f₂ = 2520 → 3f₁+7f₂ = 252.

With f₁+f₂=52: f₁=52−f₂. 3(52−f₂)+7f₂=252 → 156+4f₂=252 → f₂=24, f₁=28.

f₁=28, f₂=24.

Solution

dᵢ = xᵢ−30: −20,−10,0,10,20. fᵢdᵢ: −120,−110,0,70,100. Σfᵢdᵢ = −60. N = 50.

Mean = 30 + (−60/50) = 30−1.2 = 28.8.

Recheck: xᵢ: 10,20,30,40,50. dᵢ: −20,−10,0,10,20. fᵢdᵢ: −120,−110,0,70,100. Sum = −60. Mean = 30−60/50 = 28.8.

Solution

N=50, n/2=25. CF: 12, 26, 34, 40, 50. Median class: 120-140 (CF=26 ≥ 25).

Median = 120 + (25−12)/14 × 20 = 120 + 13×20/14 = 120 + 18.57 = 138.57.

Solution

Modal class: 3000-3500 (f=86). l=3000, f₁=86, f₀=60, f₂=74, h=500.

Mode = 3000 + (86−60)/(172−60−74) × 500 = 3000 + 26/38 × 500 = 3000 + 342.1 = 3342.1.

Solution

Total = 91+x = 100 → x = 9. Verify: n/2 = 50. CF up to 500: 2+5+9+12+17 = 45. Median class: 500-600.

Median = 500 + (50−45)/20 × 100 = 500+25 = 525 ✓.

Solution

CF: 5, 13, 25, 34, 40. Plot: (10,5), (20,13), (30,25), (40,34), (50,40).

n/2 = 20. From the ogive, at CF = 20, x ≈ 25.8. By formula: median class 20-30, l=20, cf=13, f=12, h=10.

Median = 20 + (20−13)/12 × 10 = 20 + 5.83 = 25.83.

Solution

uᵢ: −2,−1,0,1,2. fᵢuᵢ: −14,−12,0,8,10. Σfᵢuᵢ = −8. N=50.

Mean = 50 + (−8/50)×20 = 50−3.2 = 46.8.

Solution

N=40, n/2=20. Frequencies: 5, 3, 8, 14, 10. CF: 5, 8, 16, 30, 40. Median class: 30-40 (CF=30, first ≥ 20).

l=30, cf=16, f=14, h=10. Median = 30 + (20−16)/14 × 10 = 30 + 2.86 = 32.86.

Solution

N = 40, N/2 = half of total.

The median class is the class where cumulative frequency first exceeds N/2.

Median class: 20-30

Solution

N = 62, N/2 = half of total.

The median class is the class where cumulative frequency first exceeds N/2.

Median class: 20-30

Solution

N = 56, N/2 = half of total.

The median class is the class where cumulative frequency first exceeds N/2.

Median class: 20-30

Solution

N = 55, N/2 = half of total.

The median class is the class where cumulative frequency first exceeds N/2.

Median class: 20-30

Solution

N = 44, N/2 = half of total.

The median class is the class where cumulative frequency first exceeds N/2.

Median class: 30-40

Solution

N = 70, N/2 = half of total.

The median class is the class where cumulative frequency first exceeds N/2.

Median class: 20-30

Solution

N = 50, N/2 = half of total.

The median class is the class where cumulative frequency first exceeds N/2.

Median class: 20-30

Solution

N = 48, N/2 = half of total.

The median class is the class where cumulative frequency first exceeds N/2.

Median class: 20-30

Solution

N = 64, N/2 = half of total.

The median class is the class where cumulative frequency first exceeds N/2.

Median class: 20-30

Solution

N = 60, N/2 = half of total.

The median class is the class where cumulative frequency first exceeds N/2.

Median class: 20-30

Solution

7+f₁+12+f₂+8+5 = 60 → f₁+f₂ = 28 ...(i)

xᵢ: 10,30,50,70,90,110. Σfx = 70+30f₁+600+70f₂+720+550 = 1940+30f₁+70f₂.

Mean: (1940+30f₁+70f₂)/60 = 57.6 → 30f₁+70f₂ = 3456−1940 = 1516 → 3f₁+7f₂ = 151.6.

Hmm, not integer. Let me use 57.6×60 = 3456. 30f₁+70f₂ = 1516. Dividing by 2: 15f₁+35f₂ = 758. From (i): f₁ = 28−f₂.

15(28−f₂)+35f₂ = 758 → 420+20f₂ = 758 → f₂ = 16.9. Not integer — the data set has a minor inconsistency. Standard NCERT answer: f₁ = 8, f₂ = 20 (check: 30(8)+70(20) = 240+1400 = 1640. Total Σfx = 1940+1640 = 3580. Mean = 3580/60 = 59.67). Adjusting problem: with f₁+f₂=28 and mean=57.6, exact solution is f₁≈11, f₂≈17.

Solution

Mean: xᵢ: 30,40,50,60,70. Σfx = 180+400+400+720+280 = 1980. N=40. Mean = 49.5.

Mode: Modal class 55-65 (f=12). Mode = 55 + (12−8)/(24−8−4)×10 = 55+40/12 = 58.33.

Median: n/2=20. CF: 6,16,24,36,40. Median class 45-55. Median = 45+(20−16)/8×10 = 45+5 = 50.

Check: 3(50) = 150. Mode+2Mean = 58.33+99 = 157.33. Not exact (empirical relation is approximate).

Solution

N=51, n/2=25.5. CF: 6,13,28,38,47,51. Median class: 160-165 (CF=28 first ≥ 25.5).

Median = 160 + (25.5−13)/15 × 5 = 160 + 12.5×5/15 = 160 + 4.17 = 164.17 cm.