The “Mean” is computed by adding all of the numbers in the data together and dividing by the number elements contained in the data set.Watch MEAN VIDEO tio learn more..
To calculate the arithmetic mean of a set of data, add up (sum) all of the data values (x) and then divide the result by the number of values (n).
Since ∑ is the symbol used to indicate that values are to be summed (see Sigma Notation) we obtain the following formula for the mean (¯x).
Mean of ungrouped data:
Let x1, x2, x3, …, xn be n observations then mean is obtained by dividing the sum of n observations by n. It is denoted by X’ = ∑Xi/n
For Eg: Find the mean of 4,6,8,6,7,8
Mean X’ = ∑Xi/n
= (4 + 6 + 8 + 6 + 7 + 8)/6
In most of our real life situations, data is usually so large that to make a meaningful study it needs to be condensed as grouped data. So, we need to convert given ungrouped data into grouped data and devise same method to find its mean.
The following algorithm may be used to compute Arithmetic mean by DIRECT METHOD:
- Prepare the frequency table.
- Multiply the frequency of each row with the corresponding values of variable to obtain third column containing f1x1.
- Find the sum of all entries in column 3 to obtain Σf1x1.
- Find the sum of all the frequencies in column 2 to obtain Σf1= N.
- Use the formula Mean= Σf1x1/N
Assumed mean is a guess or an assumption of the mean. Assumed mean is most commonly denoted by the letter a. It doesn’t need to be correct or even close to the actual mean and choice of the assumed mean is at your discretion except for where the question explicitly asks you to use a certain assumed mean value.
- Assumed mean is used to calculate the actual mean as well as the variance and standard deviation as we’ll see later.
- Assumed mean can be calculated from the following formula:
Step Deviation : During the application of the short-cut method for finding the mean, the deviation d, are divisible by a common number ‘h’ .In this case the di = xi – A is reduced to a great extent as di becomes di / h. So the formula of mean by this is :
Where ui = ( xi – A) / h ; h = class width and N = Σ fi
Finding mean by using this formula is known as the Step Deviation Method.Watch MEAN VIDEO tio learn more..
Mean of Grouped Data
Grouped data: In a grouped distribution, values are associated with frequencies. Grouping can be in the form of a discrete frequency distribution or a continuous frequency distribution. Whatever may be the type of distribution, cumulative frequencies have to be calculated to know the total number of items.
Cumulative frequency (cf) : Cumulative frequency of each class is the sum of the frequency of the class and the frequencies of the pervious classes, ie adding the frequencies successively, so that the last cumulative frequency gives the total number of items.
Frequency Tables : When raw data is organised it can be helpful to display it in the form of a table showing the frequency (f) with which each data item (x) occurs. Such a table is called a frequency table. However, when a larger range of data is involved it may be beneficial to first break the data down into small groups, in which case, the resulting table is referred to as a grouped frequency table.
If x1 , x2,. . ., xn are observations with respective frequencies f 1, f2 , . . ., fn , then this means observation x1 occurs f 1 times, x2 occurs f 2 times, and so on.Watch MEAN VIDEO tio learn more..
Now, the sum of the values of all the observations = f1x1 + f2x2 + . . . + fnxn, and the number of observations = f1 + f2 + . . . + f n . So, the mean x of the data is given by
Write this in short form by using the Greek letter Σ (capital sigma) which means summation. That is,
which, more briefly, is written as , if it is understood that i varies from 1 to n.
Useful for CBSE, ICSE, NCERT & International Students
Grade : 9
Lesson : Statistics
Topic: MEAN VIDEO
Visit www.oztern.com to find personalized test preparation solutions for Pre Medical – AIPMT, AIIMS, JIPMER, State, Pre Engineering – IIT JEE, JEE MAIN, BITSAT, State and Foundations – Class 6 to 10.