Arithmetic Mean

Suppose the principal of your school asks your class teacher that how was the score of your class this time? What do you think is the teacher going to do? Do you think that the teacher is going to actually read out the individual score of all the students? NO!!! What the teacher does is, the teacher will tell the average score of the class instead of saying the individual score. So the principal gets an idea regarding the performance of the students. So let us now study the topic arithmetic mean in detail.

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Arithmetic Mean

In general language arithmetic mean is same as the average of data. It is the representative value of the group of data.  Suppose we are given ‘ n ‘ number of data and we need to compute the arithmetic mean, all that we need to do is just sum up all the numbers and divide it by the total numbers.

Browse more Topics under Data Handling

• Data and its Frequency Distribution
• Pictographs
• Bar Graphs
• Histogram and Pie-Charts
• Chance and Probability
• Median and Mode

Let us understand this with an example:

There are two sisters, with different heights. The height of the younger sister is 128 cm and height of the elder sister us 150cm. So what if you want to know the average height of the two sisters? What if you are asked to find out the mean of the heights?  As their total height is divided into two equal parts,

\( \frac{128 + 150}{2} \) = \( \frac{278}{2} \) = 139 cm

So 139 cm is the average height of the sisters. Here 150 > 139 >  128. Also, the average value also lies in between the minimum value and the maximum value.

Formula for Arithmetic Mean

Mean = \( \frac{Sum of all observations}{number of observations} \)

Thus, the mean of n observation x₁, x₂, . . ., x_n, is given by

\(\overline{x} \)  = \(\frac{x_1 + x_2 +….+ x_n}{n} \)  = \(  \sum_{i=1}^{n} \) x_i

Where the symbol ∑ called sigma which stands for summation. Suppose you want to find the mean score of all the tests you gave in this month.

Marks	Frequency
2	2
5	3
7	4
8	2
9	3
10	3

To find the sum of all the scores, you have to multiply the frequency of each score, with the marks obtained.

Mean = \( \frac{Sum of all observations}{number of observations} \)

So here we get another formula,

\( \overline{x} =  \frac{∑ f_x}{∑ f} \)

Now that data may be in several forms. One such type is raw data. Do you know what is Raw Data? To understand the concept of raw data, let me explain this with an example.

Suppose you are dealing with the scores of the particular cricketer over hundred matches, So for the first match, he scores x₁ runs, for the second match he scores x₂ runs and so on up to x₁₀₀ for the hundred matches.

So this is what we call as raw data. That means we are not grouping any data but presenting it as it is. So when such kind of data is presented to you and you are supposed to find out the mean, we use the formula, according to the above data,

Mean = \(  \sum_{i=1}^{100} \) x_i  /100

This is when n is 100. In general,

Mean of  \( \overline{x}  =  \sum_{i=1}^{n} \) x_i /n

This the formula for the arithmetic mean of the raw data.

Solved Examples for You

Question: The runs scored by Sachin in 5 test matches are 140, 153, 148, 150 and 154 respectively. Find the mean.

1. 150
2. 149
3. 147
4. 148

Solution: The correct option is B. Runs scored by Sachin in 5 test matches: 140, 153, 148, and 154

Means of the runs =  \( \frac{total runs}{number of matches} \)

Mean =  \( \frac{140+153+148 + 150+154}{5} \) = \( \frac{745}{5} \) = 149

Question: Mean of a set of observations is the value which

1. Occurs most frequently
2. Divides observation into two equal parts
3. is a representative of the whole number.
4. is the sum of observations

Solution: The correct option is C. Mean is the value which is the representative of the whole number. It takes into account of all the values present in the group and averages them.

Question 3: What are the type of mean?

Answer: In central tendency mean is the most common measure that we use. Moreover, there are different type of mean such as arithmetic mean, weighted mean, geometric mean, and harmonic mean. Also, if we mention it without an adjective (as mean) then, it generally refers to the arithmetic mean.

Question 4: Define geometric mean?

Answer: Geometric mean refers to the average that indicates the central tendency or typical value of a set of numbers by using the product of their values as opposed to the arithmetic mean that we also know as mean.

Question 5: Why we use the arithmetic mean?

Answer: Arithmetic mean allows us to categorize the centre of the frequency distribution of a quantitative variable by considering all of the observation with the same weight afforded to each (in contrast to the weighted arithmetic mean).

Question 6: How do you solve arithmetic mean?

Answer: For solving arithmetic mean:

• First, add up all the numbers.
• Next, divide the total by how many there are, solving Maths problems: Finding the mean.
• After that, you can find the mean by adding pieces of data together and dividing by the number of pieces of data.