Find the median wage of the following data. If the total frequency is 525, find the values of x and y.

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Understand the Problem

The question is asking to analyze a dataset and find the median wage of workers based on provided wage categories and number of workers in each category. This involves understanding how to calculate the median from the given data.

Answer

The median wage is $94.5$.
Answer for screen readers

The median wage of the workers is $94.5$.

Steps to Solve

  1. Organize the Data

    From the table, extract the wage categories and the number of workers in each category. The data is as follows:

    Wage Category No. of Workers
    Below 60 3
    60-69 27
    70-79 57
    80-89 75
    90-99 151
    100-109 104
    110-119 34
    120-129 10
    130-139 0
    140-149 29
    150-159 0
    160 and Above 60
  2. Calculate the Cumulative Frequency

    Calculate the cumulative frequency for each wage category to determine how many workers are below each wage level. This helps identify the median wage category.

    Wage Category Cumulative Frequency
    Below 60 3
    60-69 30
    70-79 87
    80-89 162
    90-99 313
    100-109 417
    110-119 451
    120-129 461
    130-139 461
    140-149 490
    150-159 490
    160 and Above 550
  3. Find the Total Number of Workers

    Add up all the workers:

    $$ \text{Total Workers} = 3 + 27 + 57 + 75 + 151 + 104 + 34 + 10 + 0 + 29 + 0 + 60 = 550 $$

  4. Determine the Median Position

    The median position is calculated by:

    $$ \text{Median Position} = \frac{\text{Total Workers}}{2} = \frac{550}{2} = 275 $$

  5. Identify the Median Category

    Locate the cumulative frequency that first reaches or exceeds 275, which occurs in the "90-99" wage category.

    The cumulative frequency just before 275 is:

    • For "80-89": 162
    • For "90-99": 313 (this exceeds 275)

    Therefore, the median wage falls in the "90-99" category.

  6. Estimate the Median Wage

    Since "90-99" is the median category, we can take the midpoint of this range to estimate the median wage:

    $$ \text{Median Wage} = \frac{90 + 99}{2} = 94.5 $$

The median wage of the workers is $94.5$.

More Information

The median wage represents the middle value, ensuring that half the workers earn less and half earn more. This statistic is crucial for understanding wage distribution within a workforce.

Tips

  • Forgetting to calculate cumulative frequencies correctly.
  • Misidentifying the median category based on incorrect cumulative frequencies.
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