Math Problem Statement

Solve for the median for grouped data

Solve for the median for grouped data.

Solution

To calculate the median for grouped data, we need to follow these steps based on the data in the image:

Step 1: Identify the relevant columns

From the image, we have the following intervals and corresponding frequencies:

Class IntervalFrequency (f)
10-192
20-297
30-3914
40-4917.8
50-5913
60-698
70-795
80-893
90-992

Step 2: Calculate cumulative frequency

Cumulative frequency (CF) is the running total of frequencies up to each class.

Class IntervalFrequency (f)Cumulative Frequency (CF)
10-1922
20-2979
30-391423
40-4917.840.8
50-591353.8
60-69861.8
70-79566.8
80-89369.8
90-99271.8

Step 3: Determine the median class

To find the median, we first need to compute the total frequency (N): N=71.8N = 71.8 The median class is the class where the cumulative frequency exceeds N/2=71.8/2=35.9N/2 = 71.8 / 2 = 35.9.

The median class is 404940-49 because the cumulative frequency becomes 40.8 in this interval, which is the first cumulative frequency greater than 35.9.

Step 4: Apply the median formula

The formula for the median of grouped data is: Median=L+(N2Ffm)×h\text{Median} = L + \left( \frac{\frac{N}{2} - F}{f_m} \right) \times h Where:

  • LL = lower boundary of the median class = 40
  • N/2=35.9N/2 = 35.9
  • FF = cumulative frequency of the class before the median class = 23
  • fmf_m = frequency of the median class = 17.8
  • hh = class width = 10 (since the class intervals are consistent)

Step 5: Calculate the median

Substitute the values into the formula: Median=40+(35.92317.8)×10\text{Median} = 40 + \left( \frac{35.9 - 23}{17.8} \right) \times 10 Median=40+(12.917.8)×10\text{Median} = 40 + \left( \frac{12.9}{17.8} \right) \times 10 Median=40+(0.7247×10)\text{Median} = 40 + (0.7247 \times 10) Median=40+7.247\text{Median} = 40 + 7.247 Median47.25\text{Median} \approx 47.25

Thus, the median is approximately 47.25.

Would you like any additional details or explanations?


Here are 5 related questions:

  1. How do you calculate the mode for grouped data?
  2. What is the difference between median and mean in grouped data?
  3. How do you find the mean for grouped data from a frequency table?
  4. Can the class width vary across different intervals when finding the median?
  5. How is the median useful in data analysis compared to other measures of central tendency?

Tip: Always check if the class intervals are consistent when using formulas for grouped data.

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Math Problem Analysis

Mathematical Concepts

Statistics
Grouped Data
Median
Cumulative Frequency

Formulas

Median = L + ((N/2 - F) / f_m) × h

Theorems

Median for Grouped Data

Suitable Grade Level

Grades 9-12