Question

Write short notes on Harmonic Mean, Arithmetic Mean, Geometric Mean, Median, Mode, Range, Standard Deviation, Interquartile Deviation, Skewness, Kurtosis and using practical examples, explain how these measures are useful in the field of Business Administration. (10marks)

Answer 1

1)Arithmetic Mean:-

he arithmetic mean is the simplest and most widely used measure of a mean, or average. It simply involves taking the sum of a group of numbers, then dividing that sum by the count of the numbers used in the series.

For example, take 34, 44, 56 and 78. The sum is 212 and divide bye 4 we gate mean 53

2)Harmonic Mean:-

The harmonic mean H of the positive real numbers {\displaystyle x_{1},x_{2},\ldots ,x_{n}} is defined to be

{\displaystyle H={\frac {n}{{\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}}}={\frac {n}{\sum \limits _{i=1}^{n}{\frac {1}{x_{i}}}}}=\left({\frac {\sum \limits _{i=1}^{n}x_{i}^{-1}}{n}}\right)^{-1}.}

3)Geometric Mean:-

The geometric mean is a mean or average, which 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 which uses their sum). The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers x₁, x₂, ..., x_n, the geometric mean is defined as

{\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}}

4)Median

The median is the value separating the higher half from the lower half of a data sample (a population or a probability distribution). For a data set, it may be thought of as the "middle" value. For example, in the data set {1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth largest, and also the fourth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it

5)Mode

Mode is the most frequently repeated  observed call as Mode .

6)Range

Range means the difference between the lowest and highest values. In {4, 6, 9, 3, 7} the lowest value is 3, and the highest is 9, so the range is 9 − 3 = 6. Range can also mean all the output values of a function.

7)Standard Deviation

The standard deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance. It is calculated as the square root of variance by determining the variation between each data point relative to the mean. If the data points are further from the mean, there is a higher deviation within the data set; thus, the more spread out the data, the higher the standard deviation.

8)Interquartile Deviation

The interquartile range is equal to Q3 minus Q1 ( i.e Q3 - Q1). For example, consider the following numbers: 1, 3, 4, 5, 5, 6, 7, 11. Q1 is the middle value in the first half of the data set. ... The interquartile range is Q3 minus Q1, so IQR = 6.5 - 3.5 = 3.

9)Skewness,

Skewness refers to distortion or asymmetry in a symmetrical bell curve, or normal distribution, in a set of data. If the curve is shifted to the left or to the right, it is said to be skewed. Skewness can be quantified as a representation of the extent to which a given distribution varies from a normal distribution. A normal distribution has a skew of zero, while a lognormal distribution, for example, would exhibit some degree of right-skew.

10)Kurtosis

Kurtosis is like skewness, kurtosis is a statistical measure that is used to describe the distribution. Whereas skewness differentiates extreme values in one versus the other tail, kurtosis measures extreme values in either tail. Distributions with large kurtosis exhibit tail data exceeding the tails of the normal distribution (e.g., five or more standard deviations from the mean). Distributions with low kurtosis exhibit tail data that are generally less extreme than the tails of the normal distribution.

Thank you..!!