How to Calculate Mean Deviation in Continuous Data
In Statistics, data representation methods can be of any type such as graphical, tabular, etc. all these represent the changes in the data maintaining some particular parameters. From the data representation, we can get the frequency of the real data. Here, frequency means the number of times of observation within a particular time interval. This frequency indication in the data representation is called a frequency distribution.
Suppose the data of the representation table are in a continuous order and maintain a particular frequency that is called a continuous frequency distribution. Mean deviation for grounded data is one of the most important information of the continuous frequency distribution along with standard deviation.
Mean Deviation and its Coefficient
The mean deviation is defined as a statistical measure used to calculate the mean of the absolute deviations of observations from some suitable average which such as the arithmetic mean, the median or the mode. It is also called the average deviation.
The difference (X–average)(X–average) is called deviation, and when the negative sign is ignored, this deviation is written as |X–average||X–average.| It is referred to as the mod deviations. The mean of mod or absolute deviations is defined as the mean deviation or the mean absolute deviation.
Therefore, for sample data in which the suitable average is the x, the mean deviation M.D. is given by the relation:
Mean:
Coefficient of M.D = Mean Deviation from Mean/Mean
Coefficient of M.D (about mean) = Mean Deviation from Mean/Mean
Median:
Coefficient of M.D = Mean Deviation from Median/Median
Coefficient of M.D (about median) = Mean Deviation from Median/Median
Mode:
Coefficient of M.D = Mean Deviation from Mode/Mode
Mean Deviation For Grouped Data
In the case of huge data, we need to divide them into some groups. Here comes the concept of grouped data. The entire data is divided into groups maintaining a particular interval. The class intervals are in a way so that no unnecessary gap comes and respective frequency is maintained. To understand the concept of grouped data, here is an example. The given table represents the information of corona infected people and their age groups in a certain town.
Example for Mean Deviation For Grouped Data
From this tabular representation, it is clear that the groups maintain a certain frequency according to the interval, and the data is continuous. We can find out the mean deviation for grouped data from such data representation using the mean deviation formula for grouped data.
Mean Deviation of Continuous Frequency Distribution
To find out the mean deviation of a continuous frequency distribution, you have to follow some steps, and they are:
Step 1: Assume the entire group is centred at a mid-point of the group. Find out the midpoint of the class. From the above example, we get:
The mean calculation formula is,
\[\bar{x}=\frac{(\sum_{i=1}^{n}x_if_i)}{(N)}\]
Step 2: The mean deviation formula for continuous series is,
\[\textrm{M.A.D}(\bar{x})=\frac{\sum_{i=1}^{n}f_i\mid x_i-\bar{x}\mid }{N}\]
Now, you have to represent the next data table using the mean other information from the above table.
So, M.A.D (x̅) = ∑i=1nfi|xi-x̅|= 8325/30 = 277.5
Standard Deviation Continuous Series Formula
Standard deviation is one of the most important statistical tools for the measurement of the distribution. Mathematically, the standard deviation is the square root of the variance of the distribution. Generally, the standard deviation is denoted as sigma (σ). To find out the standard deviation, we have to follow some steps. First of all, we calculate the arithmetic mean. Then, we have to calculate the deviation for each observation using the formula, D = X - mean.
After that comes the calculation of a variance. Variance is calculated by the division of the summation of squares of these deviations by the observation numbers. Finally, we calculate the square root of the variance and the arithmetic value is the standard deviation. Therefore, the formula of the standard deviation of continuous series is,
\[\sqrt{\sum_{i=1}^{n}f_i(x_i-\bar{x})^2/N}\]
Here,
N = number of observations.
fi = frequency values.
xi = mid-point values.
x = mean of mid-point ranges.
Formula to Find Mean by Step Deviation Method
Sometimes, the process of calculating the mean deviation for grouped data becomes complicated. For those cases, we use the step deviation method to find out the mean deviation for grouped data. We assume the middle value or close to mid-value as the mean value. The formula for this method is M.A.D (x̅) = a + (h∑i=1n fidi)/N.
Where a is the assumed mean, h is the common factor, and d = (xi-a)/h.
Solved Examples
1. Find out the M.A.D for the given data.
The mean is, x̅ = (∑i=1nxifi)/N
M.A.D (x̅) = (∑i=1nfi|xi – x̅|)/N = 1090.1/133 = 8.196
Mean Deviation about Mean
The mean is an expected value of a data set. The mean is simply defined as the sum of all observations divided by the total number of observations. The formulas for mean deviation about the mean are as the following:
Ungrouped data MAD = ∑n1|xi−μ|/n
Where, mean is μ = (x1+x2+...+xn)/n
Continuous and discrete frequency distribution MAD = ∑n1fi|xi−μ|/∑ n1fi
Where mean of grouped data is μ = ∑n1xifi/∑n1fi
Mean Deviation about Median
Median is the value that separates the lower half of the data from the upper half. Given below are the various formulas for the mean deviation about the median:
Ungrouped data MAD = ∑n1|xi−M|/n
Where, if n is odd, then median M = ((n+1)/2)th observation.
if n is even, then median M = ((n/2)thobs+(n/2+1)thobs)/2
Discrete frequency distribution MAD = ∑n1fi|xi−M|/∑n1fi
The median is calculated in the same way as ungrouped data.
Continuous frequency distribution MAD = ∑n1fi|xi−M|∑n1fi
Median of grouped data M = l+(((∑n1fi)/2)−cf)/f×h
cf stands for cumulative frequency preceding the median class,
l is the lower value of the median class,
h is the length of the median class,
f is the frequency of the median class.
Mean Deviation about Mode
The simple definition of Mode is defined as the value that occurs most frequently in a given data set. The formulas to calculate mean deviation about mode are given below:
Ungrouped data MAD = (∑n1|xi−mode|)/n
Where mode = the most frequently occurring value in a data set.
Discrete frequency distribution MAD = (∑n1fi|xi−mode|)/(∑n1fi)
The mode can be calculated in the same way as ungrouped data
Continuous frequency distribution MAD = (∑n1fi|xi−mode|)/(∑n1fi)
where, mode of grouped data = l+((f−f1)/(2f−f1−f2))×h
l is the lower value of the modal class,
h is the size of the modal class,
f is the frequency of the modal class,
f1 is the frequency of the class preceding the modal class,
f2 is the frequency of the class succeeding the modal class.
Conclusion
This is all about the mean deviation in different cases and its related formulas. Consider focusing on the different terms used in every formula and their implementation to find out the mean deviation. Focus on how these formulas are used to solve the problems given here.
FAQs on Mean Deviation in Continuous Frequency Distribution
1. What is the formula to calculate mean deviation for a continuous frequency distribution?
For a continuous frequency distribution, the mean deviation can be calculated about the mean or the median. The formulas are as follows:
1. Mean Deviation about the Mean (M.D.(x̄)):
M.D.(x̄) = (Σ fᵢ |xᵢ - x̄|) / N
2. Mean Deviation about the Median (M.D.(M)):
M.D.(M) = (Σ fᵢ |xᵢ - M|) / N
Where:
xᵢ is the mid-point of each class interval.
fᵢ is the frequency of each class interval.
N is the sum of all frequencies (Σfᵢ).
x̄ is the mean of the distribution.
M is the median of the distribution.
2. What are the key steps to calculate the mean deviation about the mean for a continuous frequency distribution?
To calculate the mean deviation about the mean for a continuous frequency distribution, follow these steps as per the CBSE Class 11 syllabus:
Step 1: Find the mid-point (xᵢ) for each class interval. The mid-point is calculated as (Upper Limit + Lower Limit) / 2.
Step 2: Calculate the mean of the distribution (x̄) using the formula: x̄ = (Σfᵢxᵢ) / N, where N = Σfᵢ.
Step 3: For each class, find the absolute deviation of its mid-point from the mean, which is |xᵢ - x̄|.
Step 4: Multiply each absolute deviation by its corresponding frequency (fᵢ) to get fᵢ|xᵢ - x̄|.
Step 5: Sum up all the values from Step 4 to get Σfᵢ|xᵢ - x̄|.
Step 6: Divide this sum by the total number of observations (N) to find the mean deviation.
3. How does mean deviation differ from standard deviation for a continuous series?
While both mean deviation and standard deviation measure the dispersion or spread of data in a continuous series, they differ in their calculation and sensitivity:
Calculation Method: Mean deviation uses the absolute values of the deviations from a central point (mean, median, or mode). In contrast, standard deviation uses the square of the deviations from the mean, which gives more weight to larger deviations.
Algebraic Properties: The use of absolute values in mean deviation makes it less convenient for further algebraic treatment. Standard deviation is mathematically more robust because it avoids the absolute value function.
Central Point: Mean deviation can be calculated from the mean, median, or mode. Standard deviation is always calculated from the mean.
4. What is the importance of calculating the mean deviation of a continuous frequency distribution?
Calculating the mean deviation is important as it provides a simple and clear measure of how spread out the data points are in a distribution. It gives the average distance of each observation from the central value (like the mean or median). This is useful in fields like economics and finance to understand the volatility of a dataset. For students, it serves as a foundational concept for understanding more complex measures of dispersion like standard deviation and variance.
5. Why is the mid-point of a class interval used to calculate mean deviation for a continuous distribution?
In a continuous frequency distribution, the exact values of individual data points within a class interval are unknown. We only know how many data points fall within a certain range (e.g., 10-20). To perform calculations, we need a single representative value for each class. The mid-point (or class mark) is chosen as the best assumption for the average value of all data points within that interval. This simplifies the calculation while providing a reasonable approximation of the overall deviation in the dataset.
6. What is a key limitation of using mean deviation as a measure of dispersion?
A major limitation of mean deviation is that it ignores the algebraic signs (positive or negative) of the deviations by using their absolute values. While this simplifies the calculation by preventing the sum of deviations from always being zero, it is considered a mathematical weakness. This disregard for signs makes mean deviation unsuitable for more advanced statistical analysis where the direction of deviation is important. This is a primary reason why standard deviation, which squares the deviations to make them positive, is more commonly used in higher-level statistics.
7. Can mean deviation be calculated about the mode for a continuous frequency distribution?
Yes, technically, mean deviation can be calculated about the mode. The procedure would involve finding the mode of the continuous distribution and then calculating the average of the absolute deviations from that modal value. However, this is rarely done in practice for two main reasons:
The mode can be ill-defined or not unique, especially in complex distributions.
The mean deviation is mathematically least when calculated from the median, making the median a more efficient central value for this purpose. Therefore, mean deviation about the mean or median is standard practice as per the NCERT syllabus.
