Formula sheet: Statistics

Statistics14 min readLesson 1 of 6

1.1 Measures of dispersion A measure of central tendency tells us where the data are centred, but two data sets can have the same mean and still be very different. The runs of two batsmen may average fifty, yet one may score steadily near fifty while the other alternates between nought and a century. The amount of scatter is measured by a measure of dispersion, and the standard measures are the range, the mean deviation about a central value, and the variance with its square root, the standard deviation.

Range = x_{m a x} - x_{m i n}, M.D. (x ˉ) = \frac{1}{n} i = 1 \sum n ∣ x_{i} - x ˉ ∣, M.D. (M) = \frac{1}{n} i = 1 \sum n ∣ x_{i} - M ∣

Range and mean deviation about the mean and about the median for ungrouped data

1.2 Grouped data For a frequency distribution each observation is weighted by its frequency. In a continuous distribution each class is represented by its mid-point, and the mean is computed from those mid-points. The mean deviation is then the weighted average of the absolute deviations, and N stands for the total frequency.

x ˉ = \frac{1}{N} i = 1 \sum n f_{i} x_{i}, M.D. (x ˉ) = \frac{1}{N} i = 1 \sum n f_{i} ∣ x_{i} - x ˉ ∣, N = i = 1 \sum n f_{i}

Mean and mean deviation for a discrete or continuous frequency distribution

σ^{2} = \frac{1}{n} i = 1 \sum n (x_{i} - x ˉ)^{2}, σ^{2} = \frac{1}{N} i = 1 \sum n f_{i} (x_{i} - x ˉ)^{2}, σ = σ^{2}

Variance and standard deviation for ungrouped data and for a frequency distribution

1.3 Reading the formulas correctly Three symbols need care. The letter n counts observations in ungrouped data, while N counts the total frequency in a frequency distribution; mixing them is the commonest error. The mean x bar is the central value about which deviations are measured, and for a frequency distribution it is the weighted mean, not the simple average of the distinct values. The variance carries squared units, so the standard deviation is usually quoted alongside it. When the data are grouped, the class mark stands in for every observation in the class, and this is an approximation. These conventions decide which formula applies and keep the arithmetic consistent across the chapter.

1.4 Computing formulas The variance is also written in a form that avoids calculating every deviation. The mean of the squares minus the square of the mean gives the same number, and for grouped data the frequencies weight each term. When the numbers are large, step deviations shrink them: choose an assumed mean a and a common factor h, write d = (x - a)/h, and use the step-deviation formula for the variance.

σ^{2} = \frac{1}{N} \sum f_{i} x_{i}^{2} - (\frac{1}{N} \sum f_{i} x_{i})^{2}, σ^{2} = h^{2} [\frac{1}{N} \sum f_{i} d_{i}^{2} - (\frac{1}{N} \sum f_{i} d_{i})^{2}]

Alternative variance formula and the step-deviation form used for large numbers

x ˉ = \frac{n _{1} x ˉ _{1} + n _{2} x ˉ _{2}}{n _{1} + n _{2}}, σ^{2} = \frac{n _{1} σ _{1}^{2} + n _{2} σ _{2}^{2}}{n _{1} + n _{2}} + \frac{n _{1} n _{2} ( x ˉ _{1} - x ˉ _{2} ) ^{2}}{( n _{1} + n _{2} ) ^{2}}

Combined mean and combined variance of two data sets

Exam

The variance changes by the square of a scale factor: if every observation is multiplied by k, the variance becomes k squared times the old variance. Adding the same constant to every observation shifts the mean but leaves the variance unchanged.

in hinglish

Statistics me dispersion batata hai ki data mean ke around kitna faila hua hai. Range sabse simple hai, mean deviation absolute deviations ka average hota hai, aur variance squared deviations ka average. Standard deviation variance ka square root hai. Grouped data me mid-points use karte hain aur frequencies se weight karte hain. Bade numbers ko chhota karne ke liye step deviation method use karo: d = (x - a)/h, phir variance ka formula h squared se multiply hota hai. Agar har observation ko k se multiply karo to variance k squared guna ho jaata hai, lekin constant add karne se variance nahi badalta. Exam me pehle data ka type dekho, ungrouped, discrete ya continuous, phir formula chuno. Aur ek baat: mean deviation aur variance dono nikaalne ke baad units compare karo; variance ka unit squared hota hai, isliye final answer me standard deviation likhna zyada clear hota hai.