How to Calculate Mean Deviation in Continuous Series with Examples
Frequency Distribution is the representation of data in a tabular form or a graphical form that indicates the frequency (the number of times any given observation occurs within a given particular interval). Assuming that the data is huge, for example, if we need to analyze the marks of 100 students, then it is not practical to represent this data in random. So based on class intervals, we use the concept of ‘Grouping of Data’.
Mean Deviation of Grouped Data
In frequency distribution of grouped data of continuous type, the class intervals or groups are arranged in a manner that there are no gaps between the classes and each class in the table has its corresponding frequency. The class intervals are chosen in such a way that they need to be mutually exclusive and exhaustive.
In order to understand how this concept of continuous frequency works, look at the following table that is given below.
The following table represents the age group of teachers working in a certain store:
This above table represents the continuous frequency in nature and the frequency is mentioned according to the interval of the classes.
How to Calculate the Mean Deviation of Continuous Frequency Distribution?
The following steps that are given below will help you calculate the mean deviation for continuous frequency distribution. These steps are:
Step 1: Consider the midpoint of each class to be its frequency. Then, the mean is calculated for these points. Using the above-mentioned table as an example again, the mid-points would be:
You can calculate the mean using the formula:
\[\overline{x} = \frac{1}{N} \sum_{i=1}^{n}\]
Step 2: Find the absolute mean deviation using the formula below:
\[M.A.D (\overline{x}) = \frac{1}{N} \sum_{i=1}^{n} f_{i} | x_{i} - \overline{x}|\]
Tabulating the above formula, we get:
Now, let us find the mean absolute deviation.
\[M.A.D (\overline{x}) = \frac{1}{N} \sum_{i=1}^{n} f_{i} | x_{i} - \overline{x}| = \frac{1090.1}{133} = 8.196\]
This might be a little complex method to solve but there is also another method called the step deviation method to find the mean absolute deviation. The result that you obtain using any of these methods is always the same or something very close to it. The step deviation method is less complicated than the other method. The formula used in Step Deviation method is:
\[M.A.D (\overline{x}) = a + \frac{h}{N} \sum_{i=1}^{n} f_{i}d_{i}\]
Here,
a = assumed mean
h = common factor
d = \[\frac{x_{i} - a}{h}\]
Now, to calculate the mean deviation, we need to know the median of the given set of data using a cumulative frequency that is given as:
\[M = l + \frac{\frac{N}{2} - C}{f} \times h\]
Where,
l = Median class’ lower limit
f = Median class’ frequency
h = Median class’ width
C = Cumulative frequency of the next or preceding class
The formula used to calculate the mean deviation is:
\[M.A.D (M) = \frac{1}{N} \sum_{(i=1)}^{n} f_{i} |x_{i} - M|\]
The mean and mean deviation is calculated for the above-used example as shown below:
We know that N/2 = 16. Hence, we will pick classes 25 - 35 as the median class.
\[M = l + \frac{\frac{N}{2} - C}{f} \times h\]
\[\Rightarrow 25 + \frac{16 - 14}{7} \times 10 = 27.42\]
The mean deviation of the mean is:
\[M.A.D (M) = \frac{1}{N} \sum_{(i=1)}^{n} f_{i} |x_{i} - M| = \frac{390.32}{32} = 12.19\]
FAQs on Mean Deviation for Continuous Frequency Distribution: Step-by-Step Guide
1. What is the formula for calculating the mean deviation for a continuous frequency distribution?
To calculate the mean deviation for a continuous frequency distribution, you can use one of two main formulas, depending on whether you are measuring the deviation from the mean or the median:
Mean Deviation about the Mean (M.D.(x̄)): (1/N) * Σ fᵢ |xᵢ - x̄|, where N = Σfᵢ.
Mean Deviation about the Median (M.D.(M)): (1/N) * Σ fᵢ |xᵢ - M|, where N = Σfᵢ.
In these formulas, xᵢ is the mid-point of the class interval, fᵢ is the corresponding frequency, N is the sum of all frequencies, x̄ is the mean, and M is the median of the distribution.
2. What are the steps to calculate the mean deviation for continuous grouped data?
To find the mean deviation for continuous grouped data as per the CBSE Class 11 syllabus, follow these steps:
Step 1: Calculate the central value you will deviate from, which is either the mean (x̄) or the median (M) of the distribution.
Step 2: Determine the mid-point (xᵢ) for each class interval. The mid-point is found by adding the upper and lower limits of the class and dividing by 2.
Step 3: For each class, calculate the absolute deviation of the mid-point from the central value, i.e., |xᵢ - x̄| or |xᵢ - M|.
Step 4: Multiply each absolute deviation by its corresponding frequency (fᵢ) to get fᵢ|xᵢ - x̄| or fᵢ|xᵢ - M|.
Step 5: Sum all the values from Step 4 to get Σfᵢ|xᵢ - x̄| or Σfᵢ|xᵢ - M|.
Step 6: Divide this sum by the total number of observations (N = Σfᵢ) to get the final mean deviation.
3. Can you provide a simple example of what a continuous frequency distribution looks like?
Certainly. A continuous frequency distribution groups data into class intervals. For example, if we were measuring the marks of students in a test, the data might be presented as follows:
Marks (Class Interval): 0-10, 10-20, 20-30, 30-40
Number of Students (Frequency): 5, 8, 15, 7
Here, the 'Marks' represent the continuous class intervals, and the 'Number of Students' is the frequency (fᵢ) for each interval. To calculate mean deviation, you would first find the mid-point (xᵢ) for each interval (e.g., 5 for 0-10, 15 for 10-20, and so on).
4. Why is finding the mid-point of each class interval necessary for this calculation?
In a continuous frequency distribution, the exact values of the individual data points within a class interval are unknown. We only know how many data points fall within that range. Therefore, we need a single representative value for each interval to perform calculations like mean or mean deviation. The mid-point (or class mark) serves as the best possible assumption for the average value of all data points in that class, allowing us to proceed with the statistical calculation.
5. How does mean deviation differ from standard deviation for a continuous distribution?
The primary difference lies in how they handle the deviations from the mean.
Mean Deviation uses the absolute value of the deviations (|xᵢ - x̄|). It gives a simple average of how far each point is from the mean, treating all deviations equally.
Standard Deviation, on the other hand, uses the square of the deviations ((xᵢ - x̄)²). By squaring, it places more weight on larger deviations, making it more sensitive to outliers. It is generally considered a more robust measure of dispersion in statistics.
6. Is it always necessary to calculate the mean before finding the mean deviation?
No, it is not always necessary to use the mean. Mean deviation can be calculated about any measure of central tendency. The two most common methods are calculating the deviation from the arithmetic mean or the median. The choice depends on the data's nature. For instance, if the dataset has significant outliers, calculating mean deviation about the median is often preferred because the median is less affected by extreme values than the mean.
7. What is the real-world importance of understanding the mean deviation of a dataset?
Understanding mean deviation is important because it provides a clear and straightforward measure of consistency or variability. For instance, in finance, an investor might use it to see how much the prices of a stock deviate from its average price over a period. In quality control, a manufacturer can use it to check if the dimensions of a product are consistently close to the required specification. A smaller mean deviation implies greater consistency and reliability in the data.
