Write procedure to find mean, mode and median of both grouped and ungrouped data.

 Here's a breakdown of how to find the mean, mode, and median for both grouped and ungrouped data:

Ungrouped Data

• Mean: The mean (or average) is found by adding all the numbers in the data set and then dividing by the number of values in the set.

  • Formula: Mean = (Sum of all values) / (Number of values)

  • Example: Data set: {3, 5, 2, 7, 1}

    • Sum of values: 3 + 5 + 2 + 7 + 1 = 18
    • Number of values: 5
    • Mean: 18 / 5 = 3.6

• Mode: The mode is the value that appears most frequently in the data set. A data set can have one mode (unimodal), more than one mode (multimodal), or no mode if all values occur with the same frequency.  

  • Example 1 (Unimodal): Data set: {3, 5, 2, 7, 3, 1}

    • Mode: 3 (appears twice)

  • Example 2 (Bimodal): Data set: {3, 5, 2, 5, 3, 1}

    • Modes: 3 and 5 (both appear twice)

  • Example 3 (No mode): Data set: {3, 5, 2, 7, 1}

    • No mode (all values appear once)

• Median: The median is the middle value when the data set is sorted in ascending order. If the data set has an odd number of values, the median is the middle value. If the data set has an even number of values, the median is the average of the two middle values.  

  • Example 1 (Odd number of values): Data set: {3, 5, 2, 7, 1}

    • Sorted: {1, 2, 3, 5, 7}
    • Median: 3

  • Example 2 (Even number of values): Data set: {3, 5, 2, 7, 1, 6}

    • Sorted: {1, 2, 3, 5, 6, 7}
    • Median: (3 + 5) / 2 = 4

Grouped Data

Grouped data is organized into classes or intervals, with frequencies indicating how many data points fall within each class.

• Mean:

  • Formula: Mean = Σ (Midpoint of class × Frequency) / Σ Frequency

  • Procedure:

    1. Find the midpoint of each class interval.
    2. Multiply each midpoint by its corresponding frequency.
    3. Add up all these products (Σ (Midpoint × Frequency)).
    4. Add up all the frequencies (Σ Frequency).
    5. Divide the sum of the products by the sum of the frequencies.

• Mode:

  • Modal Class: The modal class is the class interval with the highest frequency.

  • Estimated Mode: You can estimate the mode within the modal class using the following formula:

    • Mode = L + [(f1 - f0) / ((f1 - f0) + (f1 - f2))] × h

    • where:

      • L = Lower boundary of the modal class
      • f1 = Frequency of the modal class
      • f0 = Frequency of the class preceding the modal class
      • f2 = Frequency of the class succeeding the modal class  
      • h = Class width (size of the interval)

• Median:

  • Median Class: The median class is the class interval where the cumulative frequency first exceeds or equals half of the total frequency.

  • Estimated Median: You can estimate the median within the median class using the following formula:

    • Median = L + [(n/2 - cf) / f] × h

    • where:

      • L = Lower boundary of the median class
      • n = Total number of observations (sum of all frequencies)
      • cf = Cumulative frequency of the class preceding the median class
      • f = Frequency of the median class
      • h = Class width

Important Notes

• For grouped data, the mode and median are estimations based on the assumption that the data is evenly distributed within each class interval.
• When working with grouped data, it's important to have clear class intervals and their corresponding frequencies.

Remember, practice is key to mastering these concepts. Work through various examples with both ungrouped and grouped data to solidify your understanding.

If you have any specific examples you'd like to work through, or if you have any further questions, feel free to ask!