Statistics
🔒 Log in to trackhigh importance~3 Q in Tier 120 formulas⚡ 10 shortcuts5 subtopics
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Mean, weighted mean and combined mean
Mean
\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}=\frac{\sum x_i}{n}
Combined mean
\bar{x}=\frac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}
Weighted mean
\bar{x}=\frac{\sum w_i x_i}{\sum w_i}
Member replacement
\text{value brought in}=\text{old mean}+(\text{avg shift})\times n_{\text{new}}
Median and mode of raw data
Median (odd)
\text{median}=\left(\frac{n+1}{2}\right)\text{th value}
Median (even)
\text{median}=\frac{(\frac{n}{2})\text{th}+(\frac{n}{2}+1)\text{th}}{2}
Mode
\text{mode}=\text{most frequent value}
Empirical relation
\text{mode}=3\,\text{median}-2\,\text{mean}
Median and mode of grouped data
Grouped median
\text{median}=L+\frac{\frac{n}{2}-cf}{f}\,h
Grouped mode
\text{mode}=L+\frac{f_1-f_0}{2f_1-f_0-f_2}\,h
Cumulative frequency
cf_i=f_1+f_2+\cdots+f_i
Grouped mean (midpoints)
\bar{x}=\frac{\sum f_i m_i}{\sum f_i},\quad m_i=\frac{L_i+U_i}{2}
Range, variance and standard deviation
Range
\text{range}=x_{\max}-x_{\min}
Variance and SD
\sigma^2=\frac{1}{n}\sum(x_i-\bar{x})^2,\quad \sigma=\sqrt{\sigma^2}
Shift-scale rule
\text{SD}(a x+b)=|a|\,\text{SD}(x)
SD of 1 to n
\sigma=\sqrt{\frac{n^2-1}{12}}
Averages of special series
First n naturals
\sum_{1}^{n}k=\frac{n(n+1)}{2},\quad \text{mean}=\frac{n+1}{2}
Squares
\sum_{1}^{n}k^2=\frac{n(n+1)(2n+1)}{6}
Cubes
\sum_{1}^{n}k^3=\left[\frac{n(n+1)}{2}\right]^2
AP mean
\text{mean of AP}=\frac{\text{first}+\text{last}}{2}