Mode in Statistics
Statistical analysis is one of the major disciplines of mathematics that provides the tools necessary for collecting data, organizing it, analyzing it, and interpreting it. A measure commonly used to indicate the number of times an individual data point occurs is called the mode.
Although there are three different types of central tendency (mean, median, and mode), mode has a clear advantage over the others in that it signifies the most common occurrence of a data point.
Whether you examine exam scores, customer preferences, shoe sizes, or sales figures, mode gives you insight into what data point occurs most frequently.
1.0What is the Mode?
The mode is the value that appears most frequently in a data set. It is a measure of central tendency, alongside the mean and median, but is unique in that it highlights the most common or popular item, score, or number.
Key Points:
- The mode can be used with nominal, ordinal, discrete, and continuous data.
- A data set can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all if all values occur with equal frequency.
- The mode is particularly useful when analyzing categorical or non-numeric data.
2.0Calculating Mode for Grouped Data
A frequency distribution in a large amount of data usually groups the data into continuous intervals of classes, e.g., 0 to 10, 10 to 20 etc. As a result it is not possible to easily determine the mode from just the raw data since the actual numerical values are not available.
When estimating the mode of a set of grouped continuous data, we must first identify which class interval has the largest number of occurrences; this will give us what is called 'modal class'. After finding modal class, we can use the method of estimation of mode of grouped data for calculating the precise value of mode of grouped data in modal class.
The Grouped Mode Formula:
Mode=l+(2f1−f0−f2f1−f0)×h
Understanding the Formula Variables:
- l: The lower limit of the modal class.
- f1: The frequency of the modal class itself.
- f0: The frequency of the class interval immediately preceding (before) the modal class.
- f2: The frequency of the class interval immediately succeeding (after) the modal class.
- h: The width/size of the class interval (calculated as $\text{Upper limit} - \text{Lower limit}$).
3.0The Empirical Relationship: Mean, Median, and Mode
In descriptive statistics, when a dataset is moderately asymmetrical (skewed), there is a fascinating empirical relationship that connects all three primary measures of central tendency. Discovered by Karl Pearson, this formula allows you to calculate any one of the measures if the other two are already known.
The Empirical Formula: Mode=3(Median)−2(Mean)
Symmetrical Distribution: In a perfectly symmetrical bell-shaped curve (normal distribution), the mean, median, and mode are all perfectly equal to one another
(Mean=Median=Mode).
- Asymmetrical Distribution: As the data tilts or skews to the left or right, these three values pull away from each other in a predictable structural pattern defined by the formula above.
Mode for Ungrouped Data
Data that isn’t organized in groups are referred to as ungrouped data. To illustrate how to calculate the mode (the most frequent value) in ungrouped data, consider an example: a garment company has produced coats for sale in various sizes, the sizes of each coat based on their frequency can be seen in the following distribution table:
4.0Important Notes and Tips on Mode:
Listed below are a few important points that help to summarize our learning on this concept of mode.
- It is possible for the mean, median, and mode of a data set to equal each other, but that is not always the case.
- The mode is a useful way to calculate categorical data. If there are no repeatable numbers in the data set, it will not contain a mode.
- The mode for a data set that does not consist of any actual numbers can still be determined.
- When ordering the values in an ascending manner, calculating the mode is easier.
- To find the mode of ungrouped data, a simple observation can be used; while to find the mode of grouped data, there are formulas that must be followed.
Solved Examples
Question 1: Find the mode. 19, 1, 17, 19, 3, 20, 5, 2.
Answer: The mode of 19, 1, 17, 19, 3, 20, 5, 2 is 19.
To find the mode of the above data, we need to arrange them in ascending order.
Ascending order = 1, 2, 3, 5, 17, 19, 19, 20.
Now, as we can see that 19 is the most repetitive value.Thus, the mode of the above data set is 19.
Question 2: Using the empirical formula of mean median mode formula find the mean of the first five natural numbers, using the mean formula.
Solution:
The first five natural numbers = 1, 2, 3, 4, 5
Using the mean median, and mode formula
Mean = {Sum of Observations} ÷ {Total numbers of Observations}
Mean = (1 + 2 + 3 + 4 + 5) ÷ 5 = 15/5 = 3
Answer: The mean of the first five natural numbers {1, 2, 3, 4, 5} is 3.