Frequency Tables and Grade Marks

Q: 📊 What is a frequency table and what types of frequencies should I include?

A frequency table is an organized arrangement of statistical data that makes it easy to analyze and interpret information. There are several types of frequencies to include: <ul> <li class="font-claude-response-body break-words whitespace-normal" dir="ltr">absolute frequency (the count of how many times a value appears),</li> <li class="font-claude-response-body break-words whitespace-normal" dir="ltr">relative frequency (the proportion expressed as a decimal or percentage),</li> <li class="font-claude-response-body break-words whitespace-normal" dir="ltr">cumulative frequency (the running total of absolute frequencies), and</li> <li class="font-claude-response-body break-words whitespace-normal" dir="ltr">cumulative relative frequency (the running total of relative frequencies). </li> </ul> These different frequency types provide a complete picture of your data distribution and are fundamental to statistical analysis.

Definition of Frequency Distribution

The frequency distribution or frequency table is an arrangement in table form of statistical data, assigning to each data point its corresponding frequency.

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Types of Frequencies

Absolute Frequency

The absolute frequency is the number of times a certain value appears in a statistical study.

When flipping a coin $\text{[math]}$ times, $\text{[math]}$ heads come up.

It is represented by $\text{[math]}$ , although other authors represent it as $\text{[math]}$ .

$\text{[math]}$

The sum of the absolute frequencies equals the total number of data points, which is represented by $\text{[math]}$ .

$\text{[math]}$

To indicate these sums briefly, the Greek letter $\text{[math]}$ (capital sigma) is used, which is read as "sum" or "summation."

$\text{[math]}$

Relative Frequency

The relative frequency is the quotient between the absolute frequency of a certain value and the total number of data points.

It can be expressed as a percentage and is represented by $\text{[math]}$ .

$\text{[math]}$

The relative frequency is a number between $\text{[math]}$ and $\text{[math]}$ .

The sum of the relative frequencies equals $\text{[math]}$ .

\displaystyle n heads= \frac {35}{50}

\displaystyle n tails= \frac {15}{50}

\displaystyle n heads+n tails= \frac {35}{50}+ \frac {15}{50}= \frac {50}{50}=1

Cumulative Frequency

The cumulative frequency is the sum of the absolute frequencies of all values less than or equal to the value considered.

It is represented by $\text{[math]}$ .

Cumulative Relative Frequency

The cumulative relative frequency is the quotient between the cumulative frequency of a certain value and the total number of data points.

It can be expressed as a percentage.

Example:

During the month of Novemberm, the following maximum temperatures were recorded in a city in Fahrenheit:

$\text{[math]}$
$\text{[math]}$ .

In the first column of the table, we place the variable ordered from smallest to largest.

In the second column, we make the tally.

In the third column, we write the absolute frequency.

In the fourth column, we write the cumulative frequency:

In the first box, we place the first absolute frequency: $\text{[math]}$

In the second box, we add the value of the previous cumulative frequency plus the corresponding absolute frequency:

$\text{[math]}$

In the third box, we add the value of the previous cumulative frequency plus the corresponding absolute frequency:

F_2+f_3=3+6=9

The last one must equal $\text{[math]}$ (the sum of $\text{[math]}$ ).

F_8=N=31

In the fifth column, we arrange the relative frequencies $\text{[math]}$ , which are the result of dividing each absolute frequency by $\text{[math]}$ .

In the sixth column, we write the cumulative relative frequency $\text{[math]}$ .

In the first box, we place the first cumulative relative frequency.

In the second box, we add the value of the previous cumulative relative frequency plus the corresponding relative frequency, and so on until the last one, which must equal $\text{[math]}$ .

$\text{[math]}$	Count	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
27	I	1	1	0.032	0.032
28	II	2	3	0.065	0.097
29		6	9	0.194	0.290
30		7	16	0.226	0.516
31		8	24	0.258	0.774
32	III	3	27	0.097	0.871
33	III	3	30	0.097	0.968
34	I	1	31	0.032	1
		31		1

This type of frequency table is used with discrete variables.

Grouped Frequency Distribution

The grouped frequency distribution or table with grouped data is used if the variables take a large number of values or the variable is continuous.

The values are grouped into intervals with the same amplitude called classes. Each class is assigned its corresponding frequency.

Class Limits

Each class is defined by the lower class limit and the upper class limit.

Class Width

The class width is the difference between the upper and lower limits of the class.

Class Mark

The class mark is the midpoint of each interval and is the value that represents the entire interval for the calculation of certain parameters.

The class mark is represented by $\text{[math]}$ .

grouped frequency distribution graphical example

Construction of a Grouped Data Table

$\text{[math]}$ .

1st. Locate the smallest and largest values in the distribution. In this case, they are $\text{[math]}$ and $\text{[math]}$ .

2nd. Subtract them and find an integer a little larger than the difference and that is divisible by the number of intervals we want to establish.

It is advisable that the number of intervals be between $\text{[math]}$ and $\text{[math]}$ .

In this case, $\text{[math]}$ . We increase the number to $\text{[math]}$ , $\text{[math]}$ intervals.

Form the intervals keeping in mind that the lower limit of a class belongs to the interval, but the upper limit does not belong to that interval—it is counted in the next interval.

$\text{[math]}$ is the class mark, which is the midpoint of each interval.

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	2.5	1	1	0.025	0.025
$\text{[math]}$	7.5	1	2	0.025	0.050
$\text{[math]}$	12.5	3	5	0.075	0.125
$\text{[math]}$	17.5	3	8	0.075	0.200
$\text{[math]}$	22.5	3	11	0.075	0.275
$\text{[math]}$	27.5	6	17	0.150	0.425
$\text{[math]}$	32.5	7	24	0.175	0.600
$\text{[math]}$	37.5	10	34	0.250	0.850
$\text{[math]}$	42.5	4	38	0.100	0.950
$\text{[math]}$	47.5	2	40	0.050	1
		40		1

If you still have doubts about the construction of frequency tables, search for a private mathematics tutor on Superprof. We will be delighted to help you!

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Frequently asked questions

📊 What is a frequency table and what types of frequencies should I include?

A frequency table is an organized arrangement of statistical data that makes it easy to analyze and interpret information. There are several types of frequencies to include:

absolute frequency (the count of how many times a value appears),
relative frequency (the proportion expressed as a decimal or percentage),
cumulative frequency (the running total of absolute frequencies), and
cumulative relative frequency (the running total of relative frequencies).

These different frequency types provide a complete picture of your data distribution and are fundamental to statistical analysis.

🗂️ When should I use grouped data versus ungrouped data in a frequency table?

Use ungrouped data (discrete variables) when you have a small number of distinct values—each value gets its own row in the frequency table. Use grouped data (classes or intervals) when your variable is continuous or takes on a large number of different values; group these values into intervals of equal width (typically between 6 and 15 classes). For each class, calculate the class mark (the midpoint of the interval), assign the appropriate frequencies, and proceed with your statistical analysis. This grouping simplifies the data and makes patterns easier to identify.

Chapters

Types of Frequencies
Grouped Frequency Distribution
Construction of a Grouped Data Table

Construction of Frequency Tables

Definition of Frequency Distribution

Types of Frequencies

Relative Frequency

Cumulative Frequency

Grouped Frequency Distribution

Construction of a Grouped Data Table

Solved Frequency Exercises

Exercises and Problems on Variance

Solved Exercises on Arithmetic Mean

Practice Your Math Skills with Statistics Exercises I

Understanding the Median: Definition and Calculation Methods

Percentiles: What Are They and How to Calculate Them?

Calculation of Quartiles for Grouped Data

What Is a Histogram?