Question

How do you understand and interpret mean, median, mode, standard deviation, and variance?

Answer 1

Mean

1) Arithmetic Mean (AM):-

For a given set of observations, the AM may be defined as sum of all observation divided by the number of observations.Thus if a variable A assumes n values a₁, a₂, a_{3,...................}a_n, AM of x is denoted by A, is given by;

Geometric Mean:-

For a given set of n positive observations, the geometric mean is defined as the n^th root of the product of the observations.Thus a variable x assumes n values,x₁, x₂, x_{3,...................}x_n, all the value being possitive, then the geometric mean of x is given by,

Harmonic Mean:-

For a given set of non zero observations, harmonic mean is defined as the reciprocal of arithematic mean of the reciprocals of the observations. So if avariable x assumes n non zero values x₁, x₂, x_{3,...................}x_n, the harmonic mean of x is given by;

Median

Median for a given set of observation, may be defined as the middlemost value when the observations are arranged either in an ascending or descendng order of magnitude.

median of grouped frequency distribution;

Median = l₁+{(N/2-N₁)/(N_u-N₁)}xC

Where,

l₁= lower class boundary of the median class

N= total frequency

N₁= less than cumulative frequency corresponding to l_1, ( pre median class)

N_u= less than cumulative frequency corresponding to l_2, ( post median class)

l₂= upper boundary class

C= l₂₁ = length of median class

Mode

For given set of observations mode may be defined as the value that occure maximum number of times.Thus mode is that value which hasmaximun concentration of observation arround it.

Mode = l₁ + {(f₀-f₁)/(2f_0-f_-1-f₁)}x C

Where,

  l₁ = lowest class boundary of modal class

f₀ = frequencu pof modal class

f_-1 = frequency of pre-modal class

  f₁= frequency of post-modal class

C = class length of modal class

Standard Deviation

  Standard deviation for a given set of observation is defined as the root mean square deviation when the deviations are taken from arithmetic mean of observations. If a variable x assumes n values x₁, x₂, x_{3,...................}x_n, then standard deviation (s) is given by,

  

Variances

  variance is the expectation of the squared deviation of a random variablefrom its mean. Informally, it measures how far a set of (random) numbers are spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics,

variance = (standard deviation)²