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Cumulative frequency is a statistical tool used to represent the total number of times a particular value or observation occurs in a dataset or frequency distribution. It is also known as a cumulative frequency distribution, and it provides valuable information about the spread and shape of data.

In a frequency distribution, data is organized into categories or intervals, and the number of times each category occurs is recorded. The cumulative frequency of a category is the sum of all frequencies up to and including that category. For example, suppose we have a frequency distribution of the heights of students in a class. We could create a cumulative frequency distribution by listing the heights in ascending order and adding up the number of students with heights less than or equal to each value.

Cumulative frequency is useful for several reasons. It allows us to see the proportion of data that falls within a particular range or category. For example, if we are interested in the number of students in our class who are taller than 180cm, we can use the cumulative frequency to determine the proportion of students who fall into this category. Cumulative frequency also provides insight into the shape of a distribution. If the cumulative frequency increases steadily, the distribution is likely to be uniform. If the cumulative frequency increases more rapidly at some points than others, the distribution is likely to be skewed.

Cumulative frequency can be presented in several ways, including as a table or graph. A cumulative frequency table lists each category, its frequency, and its cumulative frequency. For example, suppose we have a frequency distribution of the ages of people in a population. We could create a cumulative frequency table by listing the ages in ascending order, the frequency of each age, and the cumulative frequency up to and including each age. A cumulative frequency graph is a graphical representation of a cumulative frequency table, and it is typically used to visualize the spread and shape of data.

Cumulative frequency graphs can take several forms, including an ogive or a step polygon. An ogive is a curved line that represents the cumulative frequency of a dataset. The line begins at the bottom left corner of the graph and increases steadily until it reaches the top right corner. The slope of the line represents the rate at which the cumulative frequency increases. A step polygon is a series of connected vertical and horizontal lines that represent the cumulative frequency of a dataset. The horizontal lines represent the cumulative frequency at each category, and the vertical lines connect the lines to create a polygon.

To create a cumulative frequency graph, we first need to calculate the cumulative frequency of each category. We can do this by adding the frequency of each category to the cumulative frequency of the previous category. Once we have calculated the cumulative frequency of each category, we can plot the data on a graph. We typically plot the category value on the x-axis and the cumulative frequency on the y-axis. We can then draw the ogive or step polygon by connecting the points on the graph.

Cumulative frequency can also be used to calculate percentiles and quartiles. Percentiles divide a dataset into 100 equal parts, and quartiles divide a dataset into four equal parts. To calculate the nth percentile of a dataset, we first need to calculate the cumulative frequency of each category. We can then use the formula (n/100)*N, where N is the total number of observations in the dataset, to find the rank of the nth percentile. We can then find the value of the nth percentile by identifying the category that contains the nth rank. Quartiles can be calculated using a similar method, except we divide the dataset into four parts instead of 100.