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Calculate Arithmetic Mean From The Following Cumulative Frequency Distribution

Reviewed by Calculator Editorial Team

A cumulative frequency distribution shows the total number of observations that fall below or at each value in a dataset. Calculating the arithmetic mean from this distribution involves converting the cumulative data into individual frequencies and then computing the average.

What is a Cumulative Frequency Distribution?

A cumulative frequency distribution is a table that shows the cumulative total of frequencies up to each class interval. Unlike a regular frequency distribution, which shows the count of observations in each interval, a cumulative frequency distribution shows the running total of observations.

For example, if you have test scores grouped in intervals and you want to know how many students scored 70 or below, you would look at the cumulative frequency for that interval.

Key Characteristics

  • Shows the total number of observations up to each class interval
  • Helps identify percentiles and quartiles
  • Useful for analyzing data distribution
  • Can be represented graphically with an ogive curve

How to Calculate the Arithmetic Mean

To calculate the arithmetic mean from a cumulative frequency distribution, follow these steps:

  1. Convert the cumulative frequency distribution to a regular frequency distribution
  2. Calculate the midpoint for each class interval
  3. Multiply each midpoint by its corresponding frequency
  4. Sum all the products
  5. Divide the total by the sum of all frequencies
Arithmetic Mean = Σ (Midpoint × Frequency) / Σ Frequency

Step-by-Step Process

  1. Identify the class intervals and their cumulative frequencies
  2. Subtract consecutive cumulative frequencies to get individual frequencies
  3. Calculate the midpoint for each interval using (Lower Bound + Upper Bound) / 2
  4. Multiply each midpoint by its frequency
  5. Sum all the products and divide by the total number of observations

Remember that the first cumulative frequency represents the frequency of the first interval, while subsequent frequencies are calculated by subtracting the previous cumulative frequency from the current one.

Worked Example

Let's calculate the arithmetic mean for the following cumulative frequency distribution:

Class Interval Cumulative Frequency
10-20 5
20-30 12
30-40 25
40-50 35

Solution

  1. Convert to frequency distribution:
    • 10-20: 5
    • 20-30: 12 - 5 = 7
    • 30-40: 25 - 12 = 13
    • 40-50: 35 - 25 = 10
  2. Calculate midpoints:
    • 10-20: (10 + 20)/2 = 15
    • 20-30: (20 + 30)/2 = 25
    • 30-40: (30 + 40)/2 = 35
    • 40-50: (40 + 50)/2 = 45
  3. Calculate products:
    • 15 × 5 = 75
    • 25 × 7 = 175
    • 35 × 13 = 455
    • 45 × 10 = 450
  4. Sum of products: 75 + 175 + 455 + 450 = 1155
  5. Total frequency: 5 + 7 + 13 + 10 = 35
  6. Arithmetic Mean: 1155 / 35 = 33

Result

The arithmetic mean of this cumulative frequency distribution is 33.

FAQ

What is the difference between a frequency distribution and a cumulative frequency distribution?
A frequency distribution shows the count of observations in each class interval, while a cumulative frequency distribution shows the running total of observations up to each interval.
When would I use a cumulative frequency distribution?
You would use a cumulative frequency distribution when you need to analyze the cumulative totals of your data, such as when calculating percentiles or creating an ogive curve.
Can I calculate the mean directly from a cumulative frequency distribution?
No, you must first convert the cumulative frequencies to individual frequencies by subtracting consecutive cumulative frequencies before calculating the mean.
What if my cumulative frequency distribution has open-ended intervals?
For open-ended intervals, you can estimate the midpoint by using reasonable values for the missing bounds, but this introduces some approximation into your calculation.