# Cumulative Frequency

## Understanding Frequency and Cumulative Frequency in Statistics

Frequency is a crucial measure in statistics that shows the number of times an event or outcome occurs. It helps us analyze data and understand the occurrence of events above or below a certain value. One effective way to study frequency is by using cumulative frequency, which shows the total of individual frequencies up to and including a particular point. Let's explore further with an example:

Number of pizzas eaten in a month (x) Number of people (frequency)

• 0 3
• 1 1
• 2 4
• 3 3
• 4 2
• 5 0

This frequency table tells us the number of people who ate a specific number of pizzas in a month. For instance, 4 individuals had exactly 2 pizzas. To find out how many people had a maximum of 2 pizzas, we can use the cumulative frequency at x = 2, which is the sum of frequencies for 0, 1, and 2 pizzas. In this case, it would be 3 + 1 + 4 = 8.

## Creating a Cumulative Frequency Table

A cumulative frequency table is an effective tool for analyzing frequencies and cumulative frequencies. To create one, simply add another column to the initial frequency table and calculate the cumulative frequency for each value of x. The cumulative frequency for a certain value of x is the sum of the frequency for that value and the cumulative frequency for the previous value of x. The first cumulative frequency will be the same as the frequency.

Example cumulative frequency table:

Number of pizzas eaten in a month (x) Number of people (frequency) Cumulative frequency

• 0 3 3
• 1 1 4
• 2 4 8
• 3 3 11
• 4 2 13
• 5 0 13

Another useful tool for analyzing cumulative frequencies is a cumulative frequency graph, which shows the relationship between x values and their corresponding cumulative frequencies. Here's an example:

[Image of cumulative frequency graph]

## Grouped Frequency Distribution and Cumulative Frequency

In statistics, data is often grouped into classes that represent a continuous range of values. This is particularly helpful when dealing with frequency distributions. Let's consider the following example:

Restaurant ratings (x) Number of restaurants (frequency)

• 0.0 - 1.0 28
• 1.0 - 2.0 82
• 2.0 - 3.0 45
• 3.0 - 4.0 40
• 4.0 - 5.0 35

To obtain a cumulative frequency table from this data, we can follow the same steps as before. However, we also need to calculate the class mark, which is the middle value of each class.

## Constructing a Cumulative Frequency Graph

To create a cumulative frequency graph for this data, we use the class mark for each class. The resulting graph would look like this:

[Image of cumulative frequency graph]

Note that the lowest possible value is 0, so the graph starts at (0, 0). Here's another example using grouped frequency distribution:

Mass in grams (x) Frequency

• 50 ≤ x < 70 60
• 70 ≤ x < 90 80
• 90 ≤ x < 110 100
• 110 ≤ x < 130 120
• 130 ≤ x < 150 140

To construct the cumulative frequency table, we need to calculate the class mark and cumulative frequency.

Cumulative frequency table:

Mass in grams (x) Class Mark Frequency Cumulative frequency

• 50 ≤ x < 70 60 60
• 70 ≤ x < 90 80 140
• 90 ≤ x < 110 100 240
• 110 ≤ x < 130 120 360
• 130 ≤ x < 150 140 500

And here's the corresponding cumulative frequency graph:

[Image of cumulative frequency graph]

## Obtaining Values of Medians, Quartiles, and Percentiles from Grouped Frequency Distributions

In grouped frequency distribution, it's not always possible to calculate the exact values of medians, quartiles, and percentiles. However, using cumulative frequency graphs, we can estimate these values. Keep in mind that they are approximations and may not be exact.

Here's a step-by-step process for obtaining these values accurately:

• Step 1: Obtain the Cumulative Frequency Table
Start with a grouped frequency distribution table and convert it to a cumulative frequency table.
• Step 2: Plot the Points on a Graph
Using the upper class boundary, plot the points from the cumulative frequency table on a graph, with cumulative frequency on the y-axis.
• Step 3: Draw the Best Fit Curve
Draw an approximate best fit curve through the plotted points on the graph.

## Step 4: Estimating Values from a Cumulative Frequency Graph

When presented with a frequency distribution, the values of the median, quartiles, and percentiles can be estimated by constructing a cumulative frequency table and corresponding graph. Let's explore this process using a frequency table of apple masses in grams as an example.

Frequency Table:

Mass in grams (x)Frequency
0 ≤ x < 1020
10 ≤ x < 2050
20 ≤ x < 3080
30 ≤ x < 40100
40 ≤ x < 5020

Solution:

Construct the resulting cumulative frequency table:

Mass in grams (x)FrequencyCumulative frequency
0 ≤ x < 1020
10 ≤ x < 205020
20 ≤ x < 308080
30 ≤ x < 40100180
40 ≤ x < 5020200

Plot the points from the cumulative frequency table on a graph and draw the best fit curve:

Example best-fit cumulative frequency curve, with values for median, quartiles, and percentiles:

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