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.
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 some 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.
Age Group | Infected People |
15-25 | 25 |
25-35 | 54 |
35-45 | 34 |
45-55 | 20 |
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.
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:
Age Group | X_{i} | Infected People (f_{i}) |
15-30 | 22.5 | 2 |
30-45 | 37.5 | 5 |
45-60 | 52.5 | 11 |
60-75 | 67.5 | 9 |
75-90 | 82.5 | 3 |
The mean calculation formula is, \[\overline{x} = \frac{(\sum_{i=1}^{n} x_{i} f_{i})}{(N)}\]
Step 2: The mean deviation formula for continuous series is,
\[M.A.D (\overline{x}) = (\frac{\sum_{i=1}^{n} f_{i} |x_{i} - \overline{x}|}{N}\]
Now, you have to represent the next data table using the mean other information from the above table.
Age Group | X_{i} | Infected People (f_{i}) | f_{i}x_{i} | |x_{i}-xÌ…| | f_{i}|x_{i} - xÌ…| |
15-30 | 22.5 | 2 | 45 | 301.5 | 621 |
30-45 | 37.5 | 5 | 187.5 | 295.5 | 1477.5 |
45-60 | 52.5 | 11 | 577.5 | 280.5 | 3085.5 |
60-75 | 67.5 | 9 | 607.5 | 265.5 | 2389.5 |
75-90 | 82.5 | 3 | 247.5 | 250.5 | 751.5 |
âˆ‘f_{i} = 30 | XÌ… = 333 | âˆ‘_{i=1}^{n}f_{i}|x_{i}-xÌ…|= 8325 |
So, M.A.D (xÌ…) = (âˆ‘_{i=1}^{n}f_{i}|xiâ€“xÌ…|)/N = 8325/30 = 277.5
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 (Ïƒ). 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 arithmetic value is the standard deviation. Therefore, the formula of the standard deviation of continuous series is,
\[\sqrt{\sum_{i=1}^{n} f_{i}(x_{i} - \overline{x})^{2}/{N}}\]
Here, Â Â N = number of observations.
Â Â Â Â Â Â Â Â fi = frequency values.
Â Â Â Â Â Â Â Â xi = mid-point values.
Â Â Â Â Â Â Â x = mean of mid-point ranges.
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 (xÌ…) = a + (hâˆ‘_{i=1}^{n} f_{i}d_{i})/N.
Where a is the assumed mean, h is the common factor and d = (x_{i}-a)/h.
1. Find out the M.A.D for the given data.
Age Group | Infected People |
15-25 | 25 |
25-35 | 54 |
35-45 | 34 |
45-55 | 20 |
The mean is, xÌ… = (âˆ‘_{i=1}^{n}x_{i}f_{i})/N
Age Group | X_{i} | Infected People (f_{i}) | f_{i}x_{i} | |x_{i} - xÌ…| | f_{i}|x_{i} - xÌ…| |
15-25 | 20 | 25 | 500 | 13.684 | 324.1 |
25-35 | 30 | 54 | 1620 | 3.684 | 198.936 |
35-45 | 40 | 34 | 1360 | 6.316 | 214.744 |
45-55 | 50 | 20 | 1000 | 16.316 | 352.32 |
âˆ‘f_{i} = 133 | XÌ… = 33.684 | âˆ‘_{i=1}^{n}f_{i}|x_{i} - xÌ…|= 1090.1 |
M.A.D (xÌ…) = (âˆ‘_{i=1}^{n}f_{i}|x_{i} â€“ xÌ…|)/N = 1090.1/133 = 8.196
1. How to Find the Average Deviation of the Grouped Data?
Ans: To find out the mean average deviation of a grouped data, you have to follow some calculation steps. First of all, you have to assume that each frequency is centred to a midpoint of the group and find out the midpoint value accordingly. Now, calculate the product of the frequency and the midpoint value of each observation. The midpoint value, upper, and lower limits of groups will be decided accordingly to the frequency distribution type. After that, calculate the difference of mean value and the midpoint value and multiply that with the respective frequency. Now, put all the values on the formula and calculate the mean average deviation.
2. Explain the Concept of a Frequency Distribution.
Ans: In statistics, an entire bunch of data is represented in many forms such as tally marks, table, graphs, etc. All these methods help to calculate the other tools of statistical measurement. This data presentation contains the frequency-divided into some intervals. In the data representation, the frequency changes according to the other parameters. The data can be of two types, such as continuous or discrete. For both the cases, the entire data is divided into some small groups maintaining a certain interval. The division is called a frequency distribution. The frequency distribution is of two types depending on the data type and internal, which are continuous frequency distribution and discrete frequency distribution.
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