Statistics
🔒 Log in to trackhigh importance~3 Q in Tier 128 formulas⚡ 10 shortcuts5 subtopics
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Mean, weighted mean and combined mean
Average
\bar{x} = \frac{\text{sum of values}}{\text{count}}
Total
\text{total} = \bar{x} \times n
Combined average
\bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1+n_2}
Missing value
x = n\bar{x} - \sum(\text{known values})
New member
w = (n+1)\bar{x}_{new} - n\bar{x}_{old}
Use the same pattern for a leaving member with n-1.
Median and mode of raw data
Median (odd n)
\text{value at } \frac{n+1}{2}\text{th place}
Position in the sorted list.
Median (even n)
\frac{\text{(n/2)th} + \text{(n/2+1)th}}{2}
Empirical relation
\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}
Rough rule for mildly skewed data; rearrange for any missing one.
Median from mode and mean
\text{Median} = \frac{\text{Mode} + 2\,\text{Mean}}{3}
Transform
y = kx + c \Rightarrow \text{Med}_y = k\,\text{Med}_x + c
Median and mode of grouped data
Grouped mean
\bar{x} = \frac{\sum f x}{\sum f}
x is the midpoint of each class.
Grouped median
\text{Med} = L + \frac{\frac{n}{2} - c}{f} \times h
L: lower limit of median class; c: cf before it; f: its frequency; h: width.
Grouped mode
\text{Mode} = L + \frac{f_m - f_1}{2f_m - f_1 - f_2} \times h
f_m: modal class frequency; f_1, f_2: neighbouring frequencies.
Range, variance and standard deviation
Range
R = \text{max} - \text{min}
Variance
\sigma^2 = \frac{\sum (x-\bar{x})^2}{n}
Average of squared distances from the mean.
Standard deviation
\sigma = \sqrt{\sigma^2}
Shift and scale
\text{SD}(kx + c) = |k|\,\text{SD}(x)
Adding c changes nothing; multiplying scales the SD.
First n naturals
\sigma^2 = \frac{n^2-1}{12}
Two values
\text{SD} = \frac{|p-q|}{2}
Coefficient of variation
CV = \frac{\sigma}{\bar{x}} \times 100\%
Sum of squares
\sum x^2 = n(\bar{x}^2 + \sigma^2)
Averages of special series
Sum of first n naturals
\sum k = \frac{n(n+1)}{2}
Sum of squares
\sum k^2 = \frac{n(n+1)(2n+1)}{6}
Sum of cubes
\sum k^3 = \left[\frac{n(n+1)}{2}\right]^2
The square of the sum of the first n naturals.
Mean of first n naturals
\frac{n+1}{2}
Mean of first n odds
n
Mean of first n evens
n+1
Multiples of k
\text{sum} = \frac{kn(n+1)}{2},\; \text{mean} = \frac{k(n+1)}{2}