\displaystyle \textbf{Points to Remember}

\displaystyle \textbf{1. Mean }(\bar{x})
\displaystyle \bar{x}=\frac{\text{Sum of observations}}{\text{Number of observations}}=\frac{\sum x_i}{n}
\displaystyle \text{where }\sum x_i\text{ is the sum of observations and }n\text{ is the number of observations.}
\displaystyle \text{or }\bar{x}=\frac{x_1+x_2+x_3+\cdots+x_n}{n}
\displaystyle \Rightarrow x_1+x_2+x_3+\cdots+x_n=n\times\bar{x}

\displaystyle \textbf{2. Mean for Ungrouped Data}
\displaystyle \textbf{(i) Direct Method}
\displaystyle \text{Let }x_1,x_2,x_3,\ldots,x_n\text{ be the variates and }f_1,f_2,f_3,\ldots,f_n\text{ be their} \\ \text{corresponding frequencies.}
\displaystyle \bar{x}=\frac{f_1x_1+f_2x_2+f_3x_3+\cdots+f_nx_n}{f_1+f_2+f_3+\cdots+f_n}=\frac{\sum f_ix_i}{\sum f_i}
\displaystyle \textbf{(ii) Assumed Mean Method}
\displaystyle \bar{x}=A+\frac{\sum f_id_i}{\sum f_i}
\displaystyle \text{where }A\text{ is the assumed mean, }f_i\text{ is the frequency and }d_i=(x_i-A).
\displaystyle \textbf{(iii) Step-Deviation Method}
\displaystyle \bar{x}=A+h\left(\frac{\sum f_iu_i}{\sum f_i}\right)
\displaystyle \text{where }A\text{ is the assumed mean, }h=x_2-x_1\text{ and }u_i=\frac{x_i-A}{h}.

\displaystyle \textbf{3. Median}
\displaystyle \textbf{(i) Median for Ungrouped Data}
\displaystyle \text{Arrange the observations in ascending or descending order.}
\displaystyle \text{If }n\text{ is odd, Median }=\left(\frac{n+1}{2}\right)^{\text{th}}\text{ term.}
\displaystyle \text{If }n\text{ is even, Median }=\frac{1}{2}\left[\left(\frac{n}{2}\right)^{\text{th}}\text{ term}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{ term}\right].
\displaystyle \textbf{(ii) Median of Discrete Series}
\displaystyle \text{Arrange the observations in ascending or descending order and prepare a cumulative} \\ \text{frequency table. Let the total frequency be }n.
\displaystyle \text{If }n\text{ is odd, Median }=\left(\frac{n+1}{2}\right)^{\text{th}}\text{ term.}
\displaystyle \text{If }n\text{ is even, Median }=\frac{1}{2}\left[\left(\frac{n}{2}\right)^{\text{th}}\text{ term}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{ term}\right].

\displaystyle \textbf{Measure of Central Tendency}

\displaystyle \text{The commonly used measures of central tendency (or averages) are:}
\displaystyle \text{(i) Arithmetic Mean (AM) or simply Mean}
\displaystyle \text{(ii) Median}

\displaystyle \textbf{Arithmetic Mean of Individual Observations (Ungrouped Data)}

\displaystyle \textbf{Definition: }\text{If }x_1,x_2,x_3,\ldots,x_n\text{ are }n\text{ values of a variable }X, \\ \text{then the arithmetic mean (or simply mean) is denoted by }\overline{X}\text{ and is defined as}
\displaystyle \overline{X}=\frac{x_1+x_2+x_3+\cdots+x_n}{n}=\frac{1}{n}\sum_{i=1}^{n}x_i
\displaystyle \text{Here, the symbol }\sum_{i=1}^{n}x_i\text{ denotes the sum }x_1+x_2+x_3+\cdots+x_n.
\displaystyle \text{In other words, the arithmetic mean of a set of observations is equal to their sum} \\ \text{divided by the total number of observations.}

\displaystyle \textbf{Properties of Arithmetic Mean}

\displaystyle \textbf{Property 1: }\text{If }\overline{X}\text{ is the mean of }n\text{ observations } \\ x_1,x_2,\ldots,x_n,\text{ then }\sum_{i=1}^{n}(x_i-\overline{X})=0.
\displaystyle \text{That is, the algebraic sum of the deviations from the mean is zero.}

\displaystyle \textbf{Property 2: }\text{If }\overline{X}\text{ is the mean of }n\text{ observations }x_1,x_2,\ldots,x_n, \\ \text{ then the mean of }x_1+a,x_2+a,\ldots,x_n+a\text{ is }\overline{X}+a.
\displaystyle \text{That is, if each observation is increased by }a,\text{ the mean also increases by }a.

\displaystyle \textbf{Property 3: }\text{If }\overline{X}\text{ is the mean of }x_1,x_2,\ldots,x_n, \\ \text{ then the mean of }ax_1,ax_2,\ldots,ax_n\text{ is }a\overline{X},\text{ where }a\ne0.
\displaystyle \text{That is, if each observation is multiplied by a non-zero number }a,\text{ the mean is} \\ \text{also multiplied by }a.

\displaystyle \textbf{Property 4: }\text{If }\overline{X}\text{ is the mean of }n\text{ observations }x_1,x_2,\ldots,x_n, \\ \text{ then the mean of }\frac{x_1}{a},\frac{x_2}{a},\ldots,\frac{x_n}{a}\text{ is }\frac{\overline{X}}{a},\text{ where }a\ne0.
\displaystyle \text{That is, if each observation is divided by a non-zero number }a,\text{ the mean is also} \\ \text{divided by }a.

\displaystyle \textbf{Property 5: }\text{If }\overline{X}\text{ is the mean of }n\text{ observations }x_1,x_2,\ldots,x_n, \\ \text{ then the mean of }x_1-a,x_2-a,\ldots,x_n-a\text{ is }\overline{X}-a,
\displaystyle \text{where }a\text{ is any real number.}

\displaystyle \textbf{Median}
\displaystyle \textbf{Definition: }\text{The median of a distribution is the value of the variable that divides} \\ \text{the distribution into two equal parts.}
\displaystyle \text{It is the value such that the number of observations above it equals the number below it.}

\displaystyle \textbf{Median of Ungrouped Data (Individual Observations)}
\displaystyle \text{If the values }x_i\text{ are arranged in ascending or descending order, the middle value} \\ \text{is called the median.}

\displaystyle \textbf{Steps to Find the Median:}
\displaystyle \text{1. Arrange the observations in ascending or descending order of magnitude.}
\displaystyle \text{2. Determine the total number of observations, say }n.

\displaystyle \textbf{Case 1: When }n\textbf{ is odd}
\displaystyle \text{Median}=\text{Value of }\left(\frac{n+1}{2}\right)^{\text{th}}\text{ observation}

\displaystyle \textbf{Case 2: When }n\textbf{ is even}
\displaystyle \text{Median}=\frac{\text{Value of }\left(\frac{n}{2}\right)^{\text{th}}\text{ observation}+\text{Value of }\left(\frac{n}{2}+1\right)^{\text{th}}\text{ observation}}{2}

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