Statistics Formulas PDF

# STATISTICS ## MEASURES OF CENTRAL TENDENCY: - An average value or a central value of a distribution is the value of a variable which is representative of the entire distribution, this representative value is called the measures of central tendency. - Generally the following five measures of cen...

# STATISTICS ## MEASURES OF CENTRAL TENDENCY: - An average value or a central value of a distribution is the value of a variable which is representative of the entire distribution, this representative value is called the measures of central tendency. - Generally the following five measures of central tendency: **(a)** Mathematical average - (i) Arithmetic mean - (iii) Harmonic mean **(b)** Positional average - (i) Median - (ii) Geometric mean - (ii) Mode ## 1. ARITHMETIC MEAN: - **(i)** For ungrouped dist.: - If x1, x2, ...... x are n values of variate x then their A.M. X is defined as: * $X = \frac{x_1 + x_2 + ..... + x_n}{n}$ * $X = \frac{\sum_{i=1}^{n} X_i}{n}$ * Σx = nx * $X = \frac{ΣX_i}{n}$ - **(ii)** For ungrouped and grouped freq. dist.: - If x1, x2, .... x are values of variate with corresponding frequencies f1, f2, ... f. then their A.M. is given by: * $x = \frac{f_1x_1 +f_2x_2 +....+ fx_n}{f_1+f_2 +....+f_n}$ * $x = \frac{\sum_{i=1}^{n} f_i x_i}{N}$ * $N = \sum_{i=1}^{n} f_i$ - **(iii)** By short method: - Let d₁ = x - a - $X = a + \frac{\sum fd_i}{N}$ - where a is assumed mean - **(iv)** By step deviation method: - Let $u_i= \frac{d_i}{h} = \frac{x_i-a}{h}$ - $X = a + \frac{\sum f u_i}{N}h$ - **(v)** Weighted mean: - If w₁, W2, ...... w are the weights assigned to the values X1, X2, ..... x respectively then their weighted mean is defined as: * Weighted mean = $ \frac{W_1X_1 + W_2X_2+.....+W_nX_n}{W_1+.....+W_n}$ * Weighted mean = $ \frac{\sum_{i=1}^{n} W_i X_i}{\sum_{i=1}^{n} W_i}$ - **(vi)** Combined mean: - If ¯x₁ and ¯x₂ be the means of two groups having n₁ and n₂ terms respectively then the mean (combined mean) of their composite group is given by combined mean * Combined mean = $\frac {n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$ * Combined mean = $\frac{n_1\bar{x}_1 + n_2\bar{x}_2 +n_3\bar{x}_3+....}{n_1 + n_2+n_3 + ....}$ - **(vii)** Properties of Arithmetic mean: - Sum of deviations of variate from their A.M. is always zero i.e. Σ(x - ¯x) = 0, Σf(x - ¯x) = 0 - Sum of square of deviations of variate from their A.M. is minimum i.e. Σ(x - ¯x)² is minimum - If ¯x is the mean of variate x then A.M. of (x + λ) = ¯x + λ - A.M. of (λx) = λ¯x - A.M. of (ax + b) = a¯x + b (where λ, a, b are constant) - A.M. is independent of change of assumed mean i.e. it is not effected by any change in assumed mean. ## 2. MEDIAN: - The median of a series is the value of the middle term of the series when the values are written in ascending order. - Therefore median, divided an arranged series into two equal parts. ### Formulae of median: - **(i)** For ungrouped distribution: - Let n be the number of variate in a series then * Median = $(\frac{n+1}{2})^{th}$ term, (when n is odd) * Median = Mean of $(\frac{n}{2})^{th} $ and $(\frac{n+1}{2})^{th}$ terms, (when n is even) - **(ii)** For ungrouped freq. dist.: - First we prepare the cumulative frequency (c.f.) column and Find value of N then * Median = $(\frac{N+1}{2})^{th}$ term, (when N is odd) * Median = Mean of $(\frac{N}{2})^{th}$ and $(\frac{N+1}{2})^{th}$ terms, (when N is even) - **(iii)** For grouped freq. dist: - Prepare c.f. column and find value of N/2 then find the class which contain value of c.f. is equal or just greater to N/2, this is median class * $Median = l + \frac{\frac {N}{2} - F}{f} \times h$ - where l - lower limit of median class - f - freq. of median class - F - c.f. of the class preceeding median class - h - Class interval of median class ## 3. MODE: - In a frequency distribution the mode is the value of that variate which have the maximum frequency. ### Method for determining mode : - **(i)** For ungrouped dist.: - The value of that variate which is repeated maximum number of times - **(ii)** For ungrouped freq. dist.: - The value of that variate which have maximum frequency. - **(iii)** For grouped freq. dist.: - First we find the class which have maximum frequency, this is model calss - $Mode = l + \frac{f_0-f_1}{2f_0-f_1-f_2} \times h$ - where l- lower limit of model class - f - freq. of the model class - f₁- freq. of the class preceeding model class - f₂- freq. of the class succeeding model class - h - class interval of model class ## 4. RELATION BETWEEN MEAN, MEDIAN AND MODE: - In a moderately asymmetric distribution following relation between mean, median and mode of a distribution. It is known as imprical formula. - Mode = 3 Median – 2 Mean - Note: - (i) Median always lies between mean and mode - (ii) For a symmetric distribution the mean, median and mode are coincide. ## 5. MEASURES OF DISPERSION: - The dispersion of a statistical distribution is the measure of deviation of its values about the their average (central) value. - Generally the following measures of dispersion are commonly used. - (i) Range - (ii) Mean deviation - (iii) Variance and standard deviation - **(i)** Range: - The difference between the greatest and least values of variate of a distribution, are called the range of that distribution. - If the distribution is grouped distribution, then its range is the difference between upper limit of the maximum class and lower limit of the minimum class. - Also, coefficient of range = $\frac{difference of extreme values}{sum of extreme values}$ - **(ii)** Mean deviation (M.D.): - The mean deviation of a distribution is, the mean of absolute value of deviations of variate from their statistical average (Mean, Median, Mode). - If A is any statistical average of a distribution then mean deviation about A is defined as: - Mean deviation = $\frac{\sum_{i=1}^{n} |{x_i - A}|}{n}$ (for ungrouped dist.) - Mean deviation = $\frac{\sum_{i=1}^{n} f_i |{x_i - A}|}{N}$ (for freq. dist.) - **Note**: - is minimum when it taken about the median - Coefficient of Mean deviation = $\frac{Mean deviation}{A}$ (where A is the central tendency about which Mean deviation is taken) - **(iii)** Variance and standard deviation: - The variance of a distribution is, the mean of squares of deviation of variate from their mean. It is denoted by σ² or var(x). - The positive square root of the variance are called the standard deviation. It is denoted by σ or S.D. - Hence standard deviation = + $\sqrt{variance}$ ## Formulae for variance: - **(i)** for ungrouped dist.: - $\sigma_x^2 = \frac{\sum(x_i -\bar{x})^2}{n}$ - $\sigma_x^2 = \frac{\sum x_i^2 }{n} - (\frac{\sum x_i}{n})^2$ - $\sigma_x^2 = \frac{\sum d_i^2 }{n} - (\frac{\sum d_i}{n})^2$ , where d₁ = x - a - **(ii)** For freq. dist.: - $\sigma_x^2 = \frac{\sum f_i(x_i -\bar{x})^2}{N}$ - $\sigma_x^2 = \frac{\sum f_i x_i^2 }{N} - (\frac{\sum f_i x_i}{N})^2$ - $\sigma_x^2 = \frac{\sum f_i d_i^2 }{N} - (\frac{\sum f_i d_i}{N})^2$ - $\sigma_x^2 = h^2 [\frac{\sum fu_i^2 }{N} - (\frac{\sum fu_i}{N})^2]$ where u = $\frac{d_i}{h} $ - **(iii)** Coefficient of S.D. = $\frac{σ}{¯x}$ - Coefficient of variation = $\frac{σ}{¯x}$ x 100 (in percentage) - **Note**: σ² = σ = σ² = h²σ ## 6. MEAN SQUARE DEVIATION: - The mean square deviation of a distrubution is the mean of the square of deviations of variate from assumed mean. It is denoted by S² - Hence S² = $\frac{\sum(x_i - a)^2}{n}$ = $\frac{\sum d_i^2 }{n}$ (for ungrouped dist.) - S² = $\frac{\sum f_i(x - a)^2}{N}$ = $\frac{\sum fd_i^2 }{N}$ (for freq. dist.), where d₁ = (x - a) ## 7. RELATION BETWEEN VARIANCE AND MEAN SQUARE DEVIATION: - $\sigma^2 = \frac{\sum f_i d_i^2 }{N} - (\frac{\sum f_i d_i}{N})^2$ - $\sigma^2 = s^2- d^2$, where d = x - a = $\frac{\sum f_i d_i }{N}$ - $s^2 = σ^2 + d^2$ , $s^2 ≥ σ^2$ - Hence the variance is the minimum value of mean square deviation of a distribution ## 8. MATHEMATICAL PROPERTIES OF VARIANCE: - Var.(x + λ) = Var.(x) - Var.(λx) = λ². Var(x) - Var(ax + b) = a². Var(x) - where λ, a, b, are constant - If means of two series containing n₁, n₂ terms are ¯x₁,¯x₂, and 2 their variance's are σ₁², σ₂² respectively and their combined mean is ¯x then the variance σ² of their combined series is given by following formula - $\sigma^2= \frac{n_1(\sigma₁^2 + d₁^2)+n_2(σ₂^2+d₂^2)}{(n₁ +n₂)}$ where d₁ = ¯x₁ - ¯x, d₂ = ¯x₂ - ¯x - i.e. σ² = $\frac{n_1σ₁^2 +n_2σ₂^2}{(n₁ +n₂)}$ + $\frac{n₁n₂}{(n₁ +n₂)^2}$(¯x₁-¯x₂)