Here are the definitions of the terms you asked for, written clearly so you can copy-paste directly into an MS Word file. All mathematical expressions are given in a linear format (not in fractions or formulas with horizontal bars), as per your request.
(a) Arithmetic Mean
The Arithmetic Mean is the most commonly used measure of central tendency. It is calculated by adding all the values in a data set and dividing the total by the number of values.
Formula:
Arithmetic Mean = (Sum of all values) ÷ (Number of values)
Example:
If the values are 10, 20, 30, then
Arithmetic Mean = (10 + 20 + 30) ÷ 3 = 60 ÷ 3 = 20
It gives the average value around which the data tends to cluster.
(b) Deviation
Deviation refers to the difference between an individual data point and the mean of the data set. It shows how much a particular value differs from the average.
Formula:
Deviation = Value of observation – Arithmetic Mean
Example:
If the mean is 25 and a particular value is 30, then
Deviation = 30 – 25 = 5
A positive deviation means the value is above the mean, and a negative deviation means it is below the mean.
(c) Standard Deviation
Standard Deviation is a measure of the dispersion or spread of data values around the mean. It indicates how much the values deviate, on average, from the mean of the dataset.
Formula for population standard deviation:
Standard Deviation (σ) = Square root of [(Sum of squared deviations from the mean) ÷ (Number of values)]
Formula in linear format:
Standard Deviation = Square root of [(Σ (x – x̄)²) ÷ N]
Where:
- x = individual value
- x̄ = arithmetic mean
- N = number of observations
- Σ = summation symbol (sum of all values)
A smaller standard deviation indicates that the data points are close to the mean, while a larger standard deviation means they are more spread out.
(d) Variance
Variance is the average of the squared deviations from the mean. It measures the degree of spread in the data set.
Formula:
Variance = (Σ (x – x̄)²) ÷ N
Where:
- x = individual value
- x̄ = arithmetic mean
- N = number of values
Note:
- Standard Deviation is the square root of variance.
- Variance is expressed in square units, while standard deviation is in the same units as the original data.
(e) Geometrical Mean
Geometrical Mean is a type of average that is used especially for multiplying or compounding rates, percentages, or ratios. It is the nth root of the product of n numbers.
Formula:
Geometrical Mean = (x₁ × x₂ × x₃ × … × xₙ) raised to the power of (1 ÷ n)
Example:
If the numbers are 4, 16, and 64, then
Geometrical Mean = (4 × 16 × 64) raised to the power of (1 ÷ 3)
= 4096 raised to the power of (1 ÷ 3)
= 16
It is mostly used in situations like growth rates, interest rates, population growth, etc.