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Calculate the mode and mean for the following grouped data: Class Intervals and Frequencies (25-35: 21, 35-45: 23, 45-55: 14)

Mathematics

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Answers

2 parts
1

36.82\text{36.82}

2

35.375\text{35.375}

Step-by-step solution

6 steps

Here's how to find the mode and mean for the given data:

Part 1· 3 steps

Calculate the Mode

  1. Identify the modal class.

    The modal class is the class interval with the highest frequency. From the table, the highest frequency is 23, which corresponds to the class interval 35-45.
    So, the modal class is 35-45.

  2. Extract values for the mode formula.

    The formula for the mode of grouped data is:

    Mode=L+(f1−f02f1−f0−f2)×h\text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h

    Where:
    L=35L = 35 (lower limit of the modal class)
    h=45−35=10h = 45 - 35 = 10 (size of the class interval)
    f1=23f_1 = 23 (frequency of the modal class)
    f0=21f_0 = 21 (frequency of the class preceding the modal class, 25-35)
    f2=14f_2 = 14 (frequency of the class succeeding the modal class, 45-55)

  3. Substitute the values into the mode formula and calculate.

    Mode=35+(23−212(23)−21−14)×10\text{Mode} = 35 + \left( \frac{23 - 21}{2(23) - 21 - 14} \right) \times 10 Mode=35+(246−35)×10\text{Mode} = 35 + \left( \frac{2}{46 - 35} \right) \times 10 Mode=35+(211)×10\text{Mode} = 35 + \left( \frac{2}{11} \right) \times 10 Mode=35+2011\text{Mode} = 35 + \frac{20}{11} Mode=35+1.8181...\text{Mode} = 35 + 1.8181... Mode≈36.82\text{Mode} \approx \boxed{\text{36.82}}

Part 2

35.375\text{35.375}

Calculate the Mean

  1. Create a table to find the mid-points (xix_i) and the product of frequency and mid-point (fixif_i x_i).

    Class IntervalFrequency (fif_i)Mid-point (xix_i)fixif_i x_i
    5-1561060
    15-251120220
    25-352130630
    35-452340920
    45-551450700
    55-65560300
    Total∑fi=80\sum f_i = 80∑fixi=2830\sum f_i x_i = 2830

  2. Apply the formula for the mean.

    The formula for the mean (direct method) is:

    xˉ=∑fixi∑fi\bar{x} = \frac{\sum f_i x_i}{\sum f_i}

  3. Substitute the sums and calculate the mean.

    xˉ=283080\bar{x} = \frac{2830}{80} xˉ=2838\bar{x} = \frac{283}{8} xˉ=35.375\bar{x} = \boxed{\text{35.375}}
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