The given data points are:

Mathematics

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Answers

6 parts
i

55.1\text{55.1}

ii

50-59\text{50-59}

iii

50-59\text{50-59}

iv

16.70\text{16.70}

v

15.77\text{15.77}

vi

20.24\text{20.24}

Step-by-step solution

6 steps

To solve this problem, we first need to organize the given data into a frequency distribution table using the specified class intervals (10-19, 20-29, etc.).

The given data points are:
23, 45, 67, 34, 56, 78, 89, 12, 45, 67,
54, 32, 76, 88, 43, 21, 65, 98, 54, 37,
29, 84, 71, 63, 55, 49, 91, 33, 27, 58,
62, 74, 39, 81, 47, 53, 69, 25, 72, 44,
36, 59, 83, 66, 41, 57, 73, 52, 68, 35

There are N=50N = 50 data points.

Here is the frequency distribution table:

Class IntervalClass Mark (x)Frequency (f)fxCumulative Frequency (cf)
10-1914.5114.51
20-2924.55122.56
30-3934.57241.513
40-4944.57311.520
50-5954.59490.529
60-6964.58516.037
70-7974.56447.043
80-8984.55422.548
90-9994.52189.050
Total∑f=50\sum f = 50∑fx=2755\sum fx = 2755

Part i· 1 step

Mean

  1. Calculate the mean (xˉ\bar{x}) using the formula xˉ=∑fx∑f\bar{x} = \frac{\sum fx}{\sum f}.

    xˉ=275550=55.1\bar{x} = \frac{2755}{50} = 55.1

    The mean is 55.1\boxed{\text{55.1}}.


Part ii

50-59\text{50-59}

Median class

  1. Determine the median position and identify the median class.

    The median position is N2=502=25\frac{N}{2} = \frac{50}{2} = 25.
    Looking at the cumulative frequency column, the 25th value falls within the class where the cumulative frequency first exceeds 25. This is the 50-59 class (cf = 29).
    The median class is 50-59\boxed{\text{50-59}}.


Part iii

50-59\text{50-59}

Modal class

  1. Identify the modal class.

    The modal class is the class with the highest frequency. From the table, the highest frequency is 9, which corresponds to the 50-59 class.
    The modal class is 50-59\boxed{\text{50-59}}.


Part iv

16.70\text{16.70}

Mean deviation

  1. Calculate the mean deviation (MD) using the formula MD=∑f∣x−xˉ∣∑fMD = \frac{\sum f|x - \bar{x}|}{\sum f}.

    First, we calculate ∣x−xˉ∣|x - \bar{x}| and f∣x−xˉ∣f|x - \bar{x}| for each class. xˉ=55.1\bar{x} = 55.1.

    | Class Mark (x) | ∣x−xˉ∣|x - \bar{x}| | Frequency (f) | f∣x−xˉ∣f|x - \bar{x}| | | :------------- | :-------------- | :------------ | :--------------- | | 14.5 | 40.6 | 1 | 40.6 | | 24.5 | 30.6 | 5 | 153.0 | | 34.5 | 20.6 | 7 | 144.2 | | 44.5 | 10.6 | 7 | 74.2 | | 54.5 | 0.6 | 9 | 5.4 | | 64.5 | 9.4 | 8 | 75.2 | | 74.5 | 19.4 | 6 | 116.4 | | 84.5 | 29.4 | 5 | 147.0 | | 94.5 | 39.4 | 2 | 78.8 | | Total | | ∑f=50\sum f = 50 | ∑f∣x−xˉ∣=834.8\sum f|x - \bar{x}| = 834.8 |

    MD=834.850=16.696≈16.70MD = \frac{834.8}{50} = 16.696 \approx 16.70

    The mean deviation is 16.70\boxed{\text{16.70}}.


Part v

15.77\text{15.77}

Semi-interquartile range

  1. Calculate the semi-interquartile range (SIQR) using the formula SIQR=Q3−Q12SIQR = \frac{Q_3 - Q_1}{2}.

    First, find the first quartile (Q1Q_1) and third quartile (Q3Q_3).
    Q1Q_1 position: N4=504=12.5th\frac{N}{4} = \frac{50}{4} = 12.5^{th} value. This falls in the 30-39 class.
    Q3Q_3 position: 3N4=3×504=37.5th\frac{3N}{4} = \frac{3 \times 50}{4} = 37.5^{th} value. This falls in the 70-79 class.

    The formula for a quartile is Qk=L+(kN4−cfbfm)cQ_k = L + \left(\frac{\frac{kN}{4} - cf_b}{f_m}\right)c, where L is the lower class boundary, cfbcf_b is the cumulative frequency before the quartile class, fmf_m is the frequency of the quartile class, and c is the class width.

    For Q1Q_1:
    L=29.5L = 29.5, cfb=6cf_b = 6, fm=7f_m = 7, c=10c = 10.

    Q1=29.5+(12.5−67)×10=29.5+(6.57)×10≈29.5+9.2857=38.7857Q_1 = 29.5 + \left(\frac{12.5 - 6}{7}\right) \times 10 = 29.5 + \left(\frac{6.5}{7}\right) \times 10 \approx 29.5 + 9.2857 = 38.7857

    For Q3Q_3:
    L=69.5L = 69.5, cfb=37cf_b = 37, fm=6f_m = 6, c=10c = 10.

    Q3=69.5+(37.5−376)×10=69.5+(0.56)×10≈69.5+0.8333=70.3333Q_3 = 69.5 + \left(\frac{37.5 - 37}{6}\right) \times 10 = 69.5 + \left(\frac{0.5}{6}\right) \times 10 \approx 69.5 + 0.8333 = 70.3333

    Now, calculate SIQR:

    SIQR=Q3−Q12=70.3333−38.78572=31.54762≈15.7738SIQR = \frac{Q_3 - Q_1}{2} = \frac{70.3333 - 38.7857}{2} = \frac{31.5476}{2} \approx 15.7738

    The semi-interquartile range is 15.77\boxed{\text{15.77}}.


Part vi

20.24\text{20.24}

Standard deviation

  1. Calculate the standard deviation (s) using the formula s=∑f(x−xˉ)2∑fs = \sqrt{\frac{\sum f(x - \bar{x})^2}{\sum f}}.

    First, we calculate (x−xˉ)2(x - \bar{x})^2 and f(x−xˉ)2f(x - \bar{x})^2 for each class. xˉ=55.1\bar{x} = 55.1.

    Class Mark (x)(x−xˉ)(x - \bar{x})(x−xˉ)2(x - \bar{x})^2Frequency (f)f(x−xˉ)2f(x - \bar{x})^2
    14.5-40.61648.3611648.36
    24.5-30.6936.3654681.80
    34.5-20.6424.3672970.52
    44.5-10.6112.367786.52
    54.5-0.60.3693.24
    64.59.488.368706.88
    74.519.4376.3662258.16
    84.529.4864.3654321.80
    94.539.41552.3623104.72
    Total∑f=50\sum f = 50∑f(x−xˉ)2=20481.00\sum f(x - \bar{x})^2 = 20481.00
    s=20481.0050=409.62≈20.2391s = \sqrt{\frac{20481.00}{50}} = \sqrt{409.62} \approx 20.2391

    The standard deviation is 20.24\boxed{\text{20.24}}.

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