The term ‘Mean’ is used constantly in the field of Statistics and is one of the basic methods used to obtain a result. It is also known as arithmetic mean or the average of a given set of data. It also measures the central tendency of data. The definition of a mean for a given set of data is the average calculated for a given set of numbers or data. This is referred to as the total of all the values of data provided divided by the number of data values in total for any given set of data.
The formula denoting the mean of a given set of data is as follows:
Mean = Sum of Observations/Total number of observations
The other two statistical methods used are median and mode to obtain a result for a given set of data. The median is defined as the value present in the middle of a given set of data and the mode is the frequency with which a particular number occurs in a given set of data.
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The mean value for a given set of data is calculated in a two-step process:
The values given in the data set are added up together.
The total of the values obtained is then divided by the number of values given.
The measure of central tendencies is used to describe data clusters around a central value. The mean definition indicates a varied formula used to calculate the mean depending on the data provided. The general formula to calculate the mean is as follows:
\[Mean = \frac{\text{Sum of Given Data}}{\text{Total Number of Data}}\]
When using the Sigma (∑) notation, the mean formula is:
\[\frac{\sum_{i=1}^{n} X_{i}}{N}\]
Here,
N = it is the Total number provided in a given data set.
∑ X_{i} = Total sum of all the data values.
Find the mean for the given set of random data,
3, 5, 9, 17, 19
The given set of data contains the numbers 3, 5, 9, 17, 19
The total number of numerals given is 5
Sum of the given numbers in the data set = 3 + 5 + 9 + 17 + 19 = 53
Therefore, Mean = Sum of given data/Total number of data
\[= \frac{53}{5} = 10.6\]
Hence, the mean for the given data is 10.6.
Arithmetic Mean
The arithmetic mean is one of the foremost methods used to obtain the central tendency of a set of data. It encompasses all the values provided by the data set. It is referred to as the ratio of the total sum of given observations to the total number of observations. The arithmetic mean can be positive, negative, or zero. There are two types of Arithmetic Mean,
Simple Arithmetic Mean.
Weighted Arithmetic Mean.
The formula to calculate Arithmetic mean is as follows:
\[X = \frac{\sum_{i=1}^{n} x_{i}p_{i}}{N}\]
The arithmetic mean is easy to calculate and is rigidly defined.
Geometric Mean
The second type of Mean is the Geometric Mean (GM). It is defined as the average value signifying the set of numbers of central tendency by calculating the product of their values. Multiplication of the numbers provided and take out the nth root of the multiplied numbers.
Here, n is the total number of values.
Taking an example of two numbers in a given set of data as 4 and 2, the geometric mean is equal to. \[\sqrt{(4+2)} = \sqrt{6} = 2.5\]
The difference between the arithmetic mean and the geometric mean is the method. In the arithmetic mean, we add the numbers whereas in the geometric mean we calculate the product of the numbers.
\[\text{Geometric Mean = } \sqrt[n]{\prod_{i=1}^{n} x_{i}}\]
Harmonic Mean
This is one of the methods of central tendency used in Statistics. It is the reciprocal of the arithmetic mean for a given set of data. The Harmonic Mean is based on all values from the data set and it is defined rigidly. It also provides the weightage of the mean in terms of large or small values depending on the data set. This is applied in time and average analysis.
To calculate the harmonic mean for a given set of data, where x_{1}, x_{2}, x_{3},…, x_{n} are the individual items up to n terms, then,
\[\text{Harmonic Mean = } \frac{n}{[(\frac{1}{x_{1}}) + (\frac{1}{x_{2}}) + (\frac{1}{x_{3}}) + . . . + (\frac{1}{x_{n}})]}\]
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1.What are the Applications of the Mean?
Ans: The mean for any given set of data is obtained by taking the average of the given set. The mean is calculated based on the type of data provided and the required outcome. The mean is rigidly confined. The mean can be used in solving the Fibonacci Sequence by market technicians.
It is also used to determine the rate of cell growth and cell division in certain biological experiments. Being easy to calculate, it is used to solve complex algebra and linear transformations. It is used to calculate the annual return on a portfolio, certain growth rates in finance, and risk in insurance.
2. What are the Differences Between the Three Types of Mean Used in Central Tendencies?
Ans: The three types of mean come under the category of Pythagroian mean central tendencies. The differences between the arithmetic mean, geometric mean, and harmonic mean are as follows:
Arithmetic Mean | Geometric Mean | Harmonic Mean |
It is the average taken of a given set of data. | Multiplication of all the numbers in the given data set. | It is the reciprocal of the arithmetic mean. |
Arithmetic Mean = (Σᵢ₌₁ⁿ xᵢpᵢ)/N | Geometric Mean = ⁿ√∏ⁿᵢ₌₁ xᵢ | Harmonic Mean = n/[(1/x₁) + (1/x₂) + (1/x₃) + . . . + (1/xₙ)] |
It can be positive, negative, or zero. | It can be a decimal value. | It can be positive or negative, not zero. |