NeokaalStudy

Statistics: Practice Worksheet

Mean, mode, and median of grouped data, including missing-frequency and cumulative-frequency problems.

12 problems·20–25 min·★★★★☆
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  1. Problem 1
    grouped-mean·★★☆☆☆
    For class midpoints 5, 15, 25, 35 with frequencies [3, 3, 2, 7], find the mean.
    ▶Answer
    $\frac{71}{3} \approx 23.67$
    ▶Step-by-step solution
    1. Use class marks as representative values.
    2. Apply $\bar x=\sum f_ix_i/\sum f_i$ and solve any resulting linear equation.
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  2. Problem 2
    grouped-mean·★★★☆☆
    Class midpoints are [110, 130, 150, 170] and frequencies are [7, 4, 2, 4]. Find the mean efficiently using an assumed mean.
    ▶Answer
    $\frac{2270}{17} \approx 133.53$
    ▶Step-by-step solution
    1. Use class marks as representative values.
    2. Apply $\bar x=\sum f_ix_i/\sum f_i$ and solve any resulting linear equation.
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  3. Problem 3
    grouped-mean·★★★★☆
    Equal-width classes have midpoints [25, 35, 45, 55] and frequencies [8, 4, 4, 5]. Find the mean using step-deviation.
    ▶Answer
    $\frac{265}{7} \approx 37.86$
    ▶Step-by-step solution
    1. Use class marks as representative values.
    2. Apply $\bar x=\sum f_ix_i/\sum f_i$ and solve any resulting linear equation.
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  4. Problem 4
    grouped-mean·★★★★★
    A grouped distribution has midpoints [12, 14, 16, 18, 20, 22, 24] and frequencies $[7,6,9,13,f,5,4]$. Its mean is 18. Find $f$.
    ▶Answer
    $20$
    ▶Step-by-step solution
    1. Use class marks as representative values.
    2. Apply $\bar x=\sum f_ix_i/\sum f_i$ and solve any resulting linear equation.
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  5. Problem 5
    grouped-mode·★★☆☆☆
    The modal class is 10-20, with preceding, modal, and succeeding frequencies 6, 13, 9. Find the mode.
    ▶Answer
    $16.36$
    ▶Step-by-step solution
    1. Identify the modal class and its neighbouring frequencies.
    2. Use the grouped-mode formula, or $\text{Mode}\approx3\text{Median}-2\text{Mean}$ when requested.
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  6. Problem 6
    grouped-mode·★★★☆☆
    For equal-width grouped data, the modal class 10-20 has adjacent frequencies 3, 8, 3; the separately calculated mean is 17.4. Find the mode and compare it with the mean.
    ▶Answer
    mode $\approx 15.00$; difference $\approx -2.40$
    ▶Step-by-step solution
    1. Identify the modal class and its neighbouring frequencies.
    2. Use the grouped-mode formula, or $\text{Mode}\approx3\text{Median}-2\text{Mean}$ when requested.
    Report an issue →
  7. Problem 7
    grouped-mode·★★★★☆
    A moderately skewed distribution has mean 25 and median 28. Estimate its mode using the empirical relation.
    ▶Answer
    $34$
    ▶Step-by-step solution
    1. Identify the modal class and its neighbouring frequencies.
    2. Use the grouped-mode formula, or $\text{Mode}\approx3\text{Median}-2\text{Mean}$ when requested.
    Report an issue →
  8. Problem 8
    grouped-median·★★★★★
    The frequencies for classes 0-10, 10-20, 20-30, 30-40 are [7, 5, 3, 6]. Find the median.
    ▶Answer
    $17.00$
    ▶Step-by-step solution
    1. Form cumulative frequencies and locate the class containing $N/2$.
    2. Use $\text{Median}=l+((N/2-c_f)/f)h$; first convert cumulative or inclusive classes when needed.
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  9. Problem 9
    grouped-median·★★☆☆☆
    A less-than cumulative-frequency table at upper boundaries 10, 20, 30, 40 is [6, 9, 12, 17]. Convert it to class frequencies and find the median.
    ▶Answer
    $18.33$
    ▶Step-by-step solution
    1. Form cumulative frequencies and locate the class containing $N/2$.
    2. Use $\text{Median}=l+((N/2-c_f)/f)h$; first convert cumulative or inclusive classes when needed.
    Report an issue →
  10. Problem 10
    grouped-median·★★★☆☆
    Measurements rounded to the nearest unit fall in inclusive classes 1-10, 11-20, 21-30, 31-40 with frequencies [5, 6, 6, 7]. Convert the classes to continuous form and find the median.
    ▶Answer
    $22.17$
    ▶Step-by-step solution
    1. Form cumulative frequencies and locate the class containing $N/2$.
    2. Use $\text{Median}=l+((N/2-c_f)/f)h$; first convert cumulative or inclusive classes when needed.
    Report an issue →
  11. Problem 11
    grouped-median·★★★★☆
    For classes 0-10, 10-20, 20-30, 30-40 with frequencies [4, 7, 12, 5], calculate and compare the mean, median, and mode.
    ▶Answer
    mean $=600/28\approx21.43$, median $\approx22.50$, mode $\approx24.17$
    ▶Step-by-step solution
    1. Form cumulative frequencies and locate the class containing $N/2$.
    2. Use $\text{Median}=l+((N/2-c_f)/f)h$; first convert cumulative or inclusive classes when needed.
    Report an issue →
  12. Problem 12
    grouped-median·★★★★★
    The median of classes 0-10, 10-20, 20-30, 30-40, 40-50, 50-60 with frequencies $5,x,20,15,y,5$ is 28.5 and the total frequency is 60. Find $x$ and $y$.
    ▶Answer
    $x=8,\ y=7$
    ▶Step-by-step solution
    1. Form cumulative frequencies and locate the class containing $N/2$.
    2. Use $\text{Median}=l+((N/2-c_f)/f)h$; first convert cumulative or inclusive classes when needed.
    Report an issue →