Mean Deviation Continuous Frequency

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+(((∑n1f_i)/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|)/(∑n1f_i)

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.