Mean Deviation for Continuous Distribution

In statistics, The number of times observation is made within a given time is frequency. A frequency distribution is a name given to this frequency indication in the data format. We can also assume that the data in the representation table are continuous and have a consistent frequency. This is referred to as a continuous frequency distribution. Along with standard deviation, the mean deviation for grounded data is one of the essential pieces of information in a continuous frequency distribution. 

What is the definition of continuous distribution?

The class intervals or groups in a continuous frequency distribution are constructed so that there are no spaces or gaps between the classes and each class in the table has its frequency. Furthermore, the class intervals are selected to be mutually exclusive and exhaustive.

Continuous data is data that has the potential to be of any value. Some continuous data can alter over time. It can take any numeric number that falls inside a finite or infinite value range. Continuous data can be divided into decimals or fractions, and don’t have to be in whole numbers, i.e., it can be dramatically partitioned into smaller pieces depending on measurement accuracy.

How is the mean deviation of continuous data calculated?

When dealing with large amounts of data, we must separate them into groups. This is where the concept of grouped data comes into play. The data is divided into several groups, each with its interval. The class intervals are designed so that there is no needless break and the appropriate frequency is maintained.

You must perform specific calculations to determine the mean average deviation of grouped data. First, assume that each frequency is centered on the group’s midway and calculate the midpoint value appropriately. Next, calculate the product of each observation’s frequency and its midpoint value. The frequency distribution type will determine groups’ midpoint value, upper and lower boundaries. After that, multiply the difference between the mean and midpoint values by the respective frequency. Now, enter all the data into the formula and find the mean average deviation.

Table with a continuous frequency distribution

Frequency distribution represents data in a tabular or graphic format that specifies the frequency (number of times an observation happens within a given interval). In statistics, the frequency distribution is significant. A well-structured frequency distribution allows for a thorough examination of the information structure. As a result, the groups into which the population divides are discernible.

It is a method of displaying data in a tabular manner in which each element of the data is allocated to a frequency. The goal of statistical data representation is to organize data concisely so that data analysis is straightforward. As a result, we present the more extensive data in a tabular format known as a frequency distribution table.

Why does frequency distribute the data?

In statistics, a collection of data is represented in various ways, including tally marks, tables, graphs, and so on. All of these strategies aid in the calculation of statistical measuring instruments. The frequency has been separated into different intervals in this data presentation. The frequency varies in the data representation as the other parameters vary. There are two forms of data: continuous and discrete. The data is separated into small groups with a set interval between them in both cases. A frequency distribution is a name for the division. There are two types of frequency distributions: continuous frequency distribution and discrete frequency distribution, dependent on the data type and context.

What is the formula used to find the mean deviation of continuous data?

One of the most important statistical methods for measuring the distribution is the standard deviation. The standard deviation is the square root of the variance of the distribution in mathematics. The standard deviation is usually written as sigma (𝞼). To calculate the standard deviation, we must go through stages. To begin, we compute the arithmetic mean. Then, using the formula D = X – mean, we must calculate the deviation for each observation.

After that, a variance must be calculated. The sum of squares of these deviations is divided by the observation numbers to calculate variance. Finally, we compute the square root of the variance, and the standard deviation is the arithmetic value. As a result, the standard deviation of a continuous series is calculated as follows:

   i=1nfixi–x2 /N

Here,  

N = number of observations. 

fi = frequency values. 

xi = mid-point values.

x = mean of mid-point ranges.

Steps that need to be followed while finding the mean deviation of continuous data

The steps to compute the mean deviation for a continuous frequency distribution are as follows:

Step i) Assume that each class’s frequency is located in the middle. For these mid-points, the mean is determined.

The formula x= i=1nxi fi/N  is used to compute the mean.

Step ii) The mean absolute deviation from the mean is calculated as follows:

M.A.D x= i=1nfi|xi –x |/N

The mean is sometimes estimated using the Step Deviation Method to simplify the complexity. The assumed mean is the observation in the center or close to the median value. The end outcome is much the same. As a result of this strategy, the size of the observations is reduced, and the calculation complexity is reduced.

Let us look at this example to understand this better:

Age Group                             No of passengers

15-25                                                  25

25-35                                                  54

35-45                                                  34

45-55                                                  20

Let us look at this data, for example. It shows the age group of passengers and the number of passengers on the flight. Now we have to find the mean using the formula: x=i=1n xifi / N

                                                ∑fi = 133    X̅ = 33.684                  ∑i=1nfi|xi – x̅|= 1090.1

Now, We use the formula M.A.D x= i=1nfi|xi –x |/N to calculate the mean absolute deviation; After applying the formula, we get: 1090.1/133 = 8.196

In Conclusion:

The majority of the data we deal with in real life is organized into groups. The volume of data is typically large, and it is connected with frequencies (sometimes, we divide data items into class intervals).

Because it is based on all of a series’ observations, the absolute mean deviation is one of the most significant ways to interpret and analyze a data set. It is straightforward to calculate, no matter how complex the data is. Grounded data is data that has been arranged and classified into groupings. Continuous and discrete frequency distributions are used to group data. Individual data points are given and the frequency with which they occur in this form of data.