The mean is one of the most important and commonly used measures of central tendency. There are several types of means in mathematics. In statistics, the mean of a given set of observations is equal to the sum of all the values in a collection of data divided by the total number of values in the data. In other words, we can simply add up all the values in a data set and divide by the total number of values to find the mean. However, the general procedure and formulas vary depending on the type of data specified, grouped data or non-grouped data.
Pooled data is a data set formed by aggregating individual observations of a variable in different groups, while unpooled data is a random set of observations. Let us understand the different mean formulas and methods to find the mean of the given set of observations using examples.
| 1. | What is the mean in statistics? |
| 2. | average formula |
| 3. | How do I find the average? |
| 4. | Mean of ungrouped data |
| 5. | Mean of the grouped data |
| 6. | Media types in mathematics |
| 7. | Median FAQ |
What is the mean in statistics?
The mean is a very important statistical concept in finance and is used in various areas of finance and business valuation. Common,Median, yWayare the three statisticsMeasures of central tendencyof files
medium definition
The mean is the average or a central value calculated from aTo adjustVoncountingand is used to measure the central tendency of the data. The central tendency is the statistical measure that recognizes the entire setDataor distribution by a single value. Provides an accurate description of all data. In statistics, the mean can also be defined as the sum of all observations to the total number of observations.
⇒ Given a set of data, \( X = x_{1},x_{2}, . . . ,x_{n}\), the mean (the arithmetic mean, theAverage), denoted by x̄, is the mean of the values of n \(x_{1},x_{2}, . . . ,x_{n}\).
Middle icon:The mean is presented as bar x, x̄.
common example
Examples of the mean in real life are:
- Average runs scored by a cricketer in friendlies.
- Average price of houses in a certain area calculated by real estate agents.
average formula
The mean formula in statistics for a set is defined as the sum of the observations divided by the total number of observations. The formula for calculating the mean will be useful in solving most problems involving the mean.
Average formula for ungrouped data:
The mean formula for a given set of observations can be expressed as:
mean = (sum of observations) ÷ (total number of observations)
Average formula for grouped data:
Similarly, we have a mean formula for grouped data, which is expressed as
x̄ = Σfx/Σf
is,
- x̄ = the mean of the given data set.
- f = frequency of each class
- x = mean value of the interval of each class
Therefore, the average of all the data points is called the mean.
Example:Find the mean of the first five naturalodd numbers, using the mean formula.
Solution:
The first five natural odd numbers = 1, 3, 5, 7 and 9
use average formula
mean = {sum of observations} ÷ {total number of observations}
Media = (1 + 3 + 5 + 7 + 9) ÷ 5 = 25/5 = 5
Answer: The mean of the first five natural odd numbers {1, 3, 5, 7, 9} is 5.
How do I find the average?
The mean is the most common central tendency that we know of and use. It is also commonly used as an average. We can calculate the mean of a given data set using different methods depending on the type of data provided. Let's see how to find the mean for a few different cases.
Fall 1:Suppose there are "n" number of elements in a list. {\({x_1, x_2, x_3, … , x_n }\)}
The mean can be calculated using the formula given below,
x̄ = \((x_1, x_2, x_3, … , x_n )/n\)
o
x̄ = Σx\(_i\)/n
Fall 2:Suppose there are n elements in a list given as {\({x_1, x_2, x_3, … , x_n }\)} and the frequency of each element is {\({f_1, f_2, f_3, … , f_n }\ ) } either.
The mean can be calculated using the formula given below,
x̄ = (f\(_1\)x\(_1\) + f\(_2\)x\(_2\) + f\(_3\)x\(_3\) + . . . + f\(_n \)x\(_n\))/(f\(_1\) + f\(_2\) + f\(_3\) + . . . + f\(_n\))
o
x̄ = Σf\(_i\)x\(_i\)/Σf\(_i\)
Fall 3:If the elements of a list are written as a range, e.g. B. 10 - 20, we must first calculate the class grade.
The mean can then be calculated using the formula below, where x\(_i\) is the class grade for each item.
x̄ = Σf\(_i\)x\(_i\)/Σf\(_i\)
Let's look at these methods in detail for different cases in the following sections.
Mean of ungrouped data
Unpooled data is the raw data collected from an experiment or study. In other words, a nongrouped record is basically a list of numbers. To find the mean of the ungrouped data, we simply calculate the sum of all the observations collected and divide it by the total number of observations. Follow the steps below to find the mean of a specific data set.
- Consider the given data set whose mean is to be calculated.
- Depending on the type of information available, apply one of the following formulas.
x̄ = \((x_1, x_2, x_3, … , x_n )/n\) or x̄ = Σf\(_i\)x\(_i\)/Σf\(_i\), where x\(_1\), x\(_2\), . . ., x\(_n\) are n observations and if f\(_1\), f\(_2\), . . . f\(_n\) are the respective frequencies.
Example:The height of five students is 161", 130", 145", 156", and 162", respectively. Find the average height of the students.
Solution:Find: the average height of the students.
The height of five students = 161 inches, 130 inches, 145 inches, 156 inches, and 162 inches (given)
Sum of the heights of five students = (161 + 130 + 145 + 156 + 162) = 754
use average formula,
Mean = {sum of observations} ÷ {total number of observations} = 754/5 = 150.8
Answer: The average height of the students is 150.8 inches.
Mean of the grouped data
Grouped data is a set of specific data that has been grouped into categories. It is a data set formed by the aggregation of individual observations of a variable in groups. For a grouped data mean, a frequency distribution table is created showing the frequencies of the given data set. We can calculate the mean of the given data using the following methods:
- direct method
- Mean value method adopted
- step deviation method
Calculation of the mean by the direct method
The direct method is the easiest way to find the mean of the grouped data. The steps that can be followed to find the mean of grouped data using the direct method are given below.
- Create a table with four columns as shown below,
Column 1 - class interval.
Column 2- class sign (corresponding), denoted by x\(_i\).
Column 3- Frequencies (fyo) (associated)
Column 4- x\(_i\)f\(_i\) (corresponding product of column 2 and column 3) - Calculate the mean using the formula Mean = ∑x\(_i\)f\(_i\)/∑f\(_i\)
Calculation of the mean using the assumed mean method
We apply the assumed mean method to find the mean of a grouped data set when the direct method becomes tedious. We can follow the steps below to find the mean using the assumed mean method.
- Create a table with five columns as below,
Column 1 - class interval.
Column 2- class sign (corresponding), denoted by x\(_i\). Take the central value of the class marks as the assumed mean and denote it A.
Column 3- Calculate the corresponding deviations given as, that is, di = x\(_i\) - A
Column 4- Frequencies (fyo) (associated)
Column 5- Average of d\(_i\), using the formula, Average of d\(_i\) = ∑x\(_i\)d\(_i\)/∑d\(_i\) - Finally, find the mean by adding the assumed mean to the mean of d\(_i\).
Calculation of the mean using the step deviation method
The step offset is also known as the origin scale and offset method. We apply the step deviation method to reduce the tedious calculations required to calculate the mean of the grouped data. The steps to follow when applying the Step Deviation method are given below.
- Create a table with five columns as shown below,
Column 1 - class interval.
Column 2- class sign (corresponding), denoted by x\(_i\). Take the central value of the class marks as the assumed mean and denote it A.
Column 3- Calculate the corresponding deviations given as, that is, di = x\(_i\) - A
Column 4- Calculate the values of u\(_i\) using the formula u\(_i\) = d\(_i\)/h, where h is the class width.
Column 5- Frequencies (fyo) (associated) - Find the mean of u\(_i\) = ∑x\(_i\)u\(_i\) / ∑u\(_i\)
- Finally, compute the mean by adding the assumed mean A to the product of the class width (h) times the mean of u\(_i\).
Example: A basketball club has 100 members. The different age groups of members and the number of members in each age group are tabulated below. Calculate the average age of the club members.
| age group | Membership |
| 10-20 | 17 |
| 20-30 | 22 |
| 30-40 | 20 |
| 40-50 | 21 |
| 50-60 | 20 |
Solution:
In this case, we must first calculate the class grade for each age group.
We use the formula below and calculate the class grade for each age group.
Class grade = (upper bound + lower bound)/2
| age group | class sign | Membership |
| 10-20 | quince | 17 |
| 20-30 | 25 | 22 |
| 30-40 | 35 | 20 |
| 40-50 | 45 | 21 |
| 50-60 | 55 | 20 |
Now,
\(x_1\) = 15, \(x_2\) = 25, \(x_3\) = 35, \(x_4\) = 45, \(x_5\) = 55
\(f_1\) = 17, \(f_2\) = 22, \(f_3\) = 20, \(f_4\) = 21, \(f_5\) = 20
\(x_1f_1\) =15 × 17 = 255
\(x_2f_2\) = 25 × 22 = 550
\(x_3f_3\) = 35 × 20 = 700
\(x_1f_1\) = 45 × 21 = 945
\(x_1f_1\) = 55 × 20 = 1100
f\(_1\)x\(_1\) + f\(_2\)x\(_2\) + f\(_3\)x\(_3\) + f\(_4\)x\(_4\ ) + f\(_5\)x\(_5\) = 255 + 550 + 700 + 945 + 1100 = 3550
f\(_1\) + f\(_2\) + f\(_3\) + f\(_4\) + f\(_5\) = 17 + 22 + 20 + 21 + 20 = 100
We will use the formula given below.
x̄ = Σf\(_i\)x\(_i\)/Σf\(_i\)
Median age = 3550/100
= 35,5
The average age of the members = 35.5
Media types in mathematics
There are several types of mean in mathematics: arithmetic mean, weighted mean, geometric mean (GM), and harmonic mean (HM). When mentioned without an adjective (such as mean), the mean in statistics usually refers to the arithmetic mean. Some of the media types are briefly discussed below:
- arithmetic meaning
- weighted average
- geometric meaning
- harmonic means
arithmetic meaning
arithmetic meaningIt is often referred to as the mean or arithmetic mean, which is calculated by adding all the numbers in a given data set and then dividing by the total number of items in that data set. The general formula for finding the arithmetic mean is as follows:
x̄ = Σf\(_i\)/N
is,
- x̄ = the mean of the given data set.
- f = frequency of individual data
- N = sum of frequencies
weighted average
The weighted mean is calculated when certain values reported in a data set are more important than others. Each of the values x\(_i\) is assigned a weight w\(_i\). The general formula for finding the weighted average is as follows:
Weighted average = Σw\(_i\)x̄/Σw\(_i\)
is,
- x̄ = the mean of the given data set.
- w = corresponding weight for each observation
geometric meaning
Diegeometric meaningis defined as ntheSquare root of the product of n numbers in the given data set. The formula to find the geometric mean for a given data set, \(x_1, x_2, x_3, … , x_n \),
GM =norte√(x\(_1\) · x\(_2\) · x\(_3\) · … · x\(_n\))
harmonic means
For a given set of observations, the harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations, given by the formula,
Media armónica = 1/[Σ(1/\(x_i\))]/N = N/Σ(1/\(x_i\))
Media related topics:
- categorical data
- rank in statistics
- Average
- geometric meaning
Median FAQ
What is the mean in statistics?
One of the most important and commonly used measures of central tendency, the mean is the average, or a calculated central value, of a set of numbers.
What is the formula for the medium?
There are different formulas to find the mean of a specific data set as follows:
For ungrouped data: mean = (sum of observations) ÷ (total number of observations)
For grouped data, x̄ = Σfx/Σf
is,
- x̄ = the mean of the given data set.
- f = frequency of each class
- x = mean value of the interval of each class
What is the mean formula for grouped data?
The mean formula for finding the mean of a grouped data set can be given as: x̄ = Σfx/Σf, where x̄ is the mean of the specified data set, f is the frequency of each class, and x is the mean value of each class
What is the mean formula for ungrouped data?
The mean formula for finding the mean of an ungrouped data set can be given as follows: mean = (sum of observations) ÷ (total number of observations)
What is the difference between mean formula and median formula?
The mean formula is given as the average of all observations. It is expressed as mean = {sum of observations} ÷ {total number of observations}. While the median formula depends entirely on the number of observations (n). When the number of observations iseventhen the formula for the median is [median = ((n/2)theterm + ((n/2) + 1)theterm)/2] and if n = odd then the median formula is [median = {(n + 1)/2}theExpression].
How do you calculate the mean using the mean formula?
Given the set of 'n' observations, the mean can easily be calculated using a general mean formula, ie mean = {sum of observations} ÷ {total number of observations}.
What are the different types of media?
The different types of means in mathematics are:
- arithmetic meaning
- weighted average
- geometric meaning
- harmonic means
What is the difference between the arithmetic mean and the weighted mean?
The arithmetic mean is often referred to as the arithmetic mean or average, which is calculated by adding all the numbers in a given data set and then dividing by the total number of items in that data set, while the weighted mean is calculated when certain Values given in one record are more important than others.
What are the applications of the media in our daily life?
The mean is a statistical concept that is very important in finance and is used in various areas of finance such as business valuation.
How is the mean formula used?
The formula for the overall mean is expressed mathematically as mean = {sum of observations} ÷ {total number of observations}. Let's see an example to understand its use.
Example: Find the mean of (1, 2, 3, 4, 5, 6, 7)
Solution: Total number of observations = 7
mean = {sum of observations} ÷ {total number of observations}
Media = (1 + 2 + 3 + 4 + 5 + 6 + 7) ÷ 7 = 28/7 = 4
The mean of (1, 2, 3, 4, 5, 6, 7) is 4
What is the mean formula for n observations?
The mean formula for 'n' observations is expressed as
Mean of n observations = {sum of 'n' observations} ÷ {total number of 'n' observations}