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Cumulative Frequency Distribution

A cumulative frequency distribution is a statistical table that displays the running total of frequencies across successive class intervals. It calculates the total number of observations that fall below or above a specific boundary in a grouped dataset. This distribution is crucial for finding the median and plotting ogive curves.

Key Concepts

It is constructed by continuously adding the frequency of the current class interval to the sum of frequencies from all preceding intervals.

The 'less than' type adds frequencies from the lowest class to the highest, while the 'more than' type adds frequencies from the highest class down to the lowest.

This distribution is the foundational step required to locate the median class and calculate the median of grouped data.

Formula & Equation

Median = l + \left( \frac{\frac{n}{2} - cf}{f} \right) \times h

l = lower limit of median class, n = total number of observations, cf = cumulative frequency of the class preceding the median class, f = frequency of the median class, h = class size

Common Misconceptions

Myth: Students often substitute the cumulative frequency of the median class into the median formula.

Fact: The variable 'cf' in the median formula specifically refers to the cumulative frequency of the class immediately preceding the median class, not the median class itself.

Real World Applications

A teacher uses a less than cumulative frequency table to quickly find out how many students scored below 75 marks in a mathematics board exam.

A government agency uses it to determine the total number of households earning an annual income below a specific poverty line threshold.

Frequently Asked Questions

What is the difference between frequency and cumulative frequency?

Frequency represents the exact number of items in a single class interval, whereas cumulative frequency is the running total of frequencies up to that specific class.

How do you identify the median class using a cumulative frequency distribution?

First, find half of the total frequency (n/2). The median class is the interval whose cumulative frequency is just greater than or equal to this n/2 value.

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