# Measures of Dispersion

• • • If x1, x2, x3 ....... xn are the n values of x then its arithmetic mean
\overline{x} = \frac{\sum x_i}{n}
• If x1, x2, x3 ....... xn are the n values of x then its arithmetic mean
\overline{x} = A + \frac{\sum (x_i-A}{n})
where A is Assumed mean
• If x1, x2, ....... xn are the n values of x and its corresponding frequencies are f1, f2, ...... fn, then its arithmetic mean
\tt \overline{x} = \frac{\sum f_i \ x_i}{N}
where N = Σ fi
• If x1, x2, ....... xn are the n values of x and its corresponding frequencies are f1, f2, ...... fn, then its arithmetic mean
\tt \overline{x} = A + \frac{\sum f_i \ d_i}{N}
where di = xi −A
N = Σ fi
A = Assumed mean
• If x1, x2, ....... xn are the n values of x and its corresponding sizes n1, n2, ...... nn, then its arithmetic mean is given by
\tt \overline{x} = \frac{\sum n_i \ x_i}{\sum n_i}
• If x1, x2, ....... xn are the n values of x and its corresponding weights w1, w2, ...... wn, then its arithmetic mean is given by
\tt \overline{x} = \frac{\sum w_i \ x_i}{\sum w_i}
• AM of (ax + b) = a AM (x) + b
= a\overline{x} + b
• If x1, x2, ....... xn are the n values of x, then its geometric mean is
= \sqrt[n]{x_{1}. x_{2}. x_{3}......x_{n}}
• GM = Anti log \left(\frac{\sum_{i = 1}^{n}\log (x_i)}{n}\right)
• If x1, x2, ....... xn are the mid values of n classes and its corresponding frequencies f1, f2, f3, ..... fn.
Then GM = \sqrt[N]{x_{1}^{f_{1}}, x_{2}^{f_{2}} ...... x_{n}^{f_{n}}}
where N = Σ fi
• GM = Anti log \left(\frac{\sum_{i = 1}^{n} f_i \log (x_i)}{N}\right)
where N = Σ fi
• If G1, G2, ....... Gn are the GM of the n series of sizes n1, n2, ...... nn, respectively. Then GM of the combined series is given by
\tt = \sqrt[N]{G_{1}^{n_{1}}, G_{2}^{n_{2}}, ...... G_{n}^{n_{n}}}
where N = n1 + n2 + ...... + nn
• GM = Antilog \tt \left(\frac{\sum_{i = 1}^{n} n_i \ \log (G_i)}{N}\right)
where N = n1 + n2 + ...... + nn
• If x1, x2, x3 ....... xn are the n values of x then Harmonic mean H.M = \tt \frac{n}{\frac{1}{x_{1}} + \frac{1}{x_{2}} + ..... + \frac{1}{x_{n}}}
• If x1, x2, ....... xn are the n values of x and its corresponding frequencies are f1, f2, ...... fn, then the Harmonic mean H.M = \tt \frac{\sum_{i = 1}^{n} f_i}{\sum_{i = 1}^{n} \frac{f_i}{x_i}}
• If all the numbers are equal, then A.M = G.M = H.M
• If the numbers are different A.M ≥ G.M ≥ H.M
• The relation between A.M, G.M, and H.M is (GM)2 = (AM)(HM)
• \tt velocity = \frac{distance}{time}
d1, d2, ....... dn are the distances, t1, t2, ....... tn are the times, v1, v2, ....... vn are velocities. Then average velocity = \tt \frac{d_{1} + d_{2} + ...... +d_{n}}{\frac{d_{1}}{v_{1}} + \frac{d_{2}}{v_{2}} + ..... +\frac{d_{n}}{v_{n}}}
• If d1 = d2 = ....... = dn. Then the average velocity = \tt \frac{n}{\frac{1}{v_{1}} + \frac{1}{v_{2}} + ..... + \frac{1}{v_{n}}}
• If x1, x2, ....... xn are the n-observations. Then median = \tt \left(\frac{n + 1}{2}\right)^{th} observation, if n is odd
• If x1, x2, ....... xn are the n-observations. Then median = \tt \frac{1}{2}\left[\left(\frac{n}{2}\right)^{th} observation + \left(\frac{n}{2} + 1\right)^{th} observation\right]
• If x1, x2, x3 ....... xn are the n-values of x and its corresponding frequencies are f1, f2, ....... fn. Then median = \tt \left(\frac{N + 1}{2}\right)^{th} item, if N is odd
where N = Σ fi
• If x1, x2, x3 ....... xn are the n-values of x and its corresponding frequencies are f1, f2, ....... fn. Then median = \tt \frac{1}{2}\left[\left(\frac{N}{2}\right)^{th} item + \left(\frac{N}{2} + 1\right)^{th} item\right]
where N = Σ fi
• For Continuous frequency distribution, median = l + \left(\frac{\frac{N}{2} - F}{f} \times c\right)
l = lower bound
F = cumulative frequency just before the class
f = frequency of class
c = class size
Cumulative frequency just more than \frac{N}{2} is median class.
• For individual series, quartile
Q_{t} = \left[\frac{t(n + 1)}{4}\right]^{th} observation
• For individual series, decile D_{t} = \left[\frac{t(n + 1)}{10}\right]^{th} observation
• For individual series, persontile P_{t} = \left[\frac{t(n + 1)}{100}\right]^{th} observation
• For discrete series, Quartile Q_{t} = \left[\frac{t(N + 1)}{4}\right]^{th} item
• For discrete series decile D_{t} = \left[\frac{t(N + 1)}{10}\right]^{th} item
• For the discrete series, the persontile P_{t} = \left[\frac{t(N + 1)}{100}\right]^{th} item
• For the continuous series, the Quartile Q_{t} = l + \left(\frac{\frac{tN}{4} - F}{f} \times c\right)
• For the continuous series, the decile D_{t} = l + \left(\frac{\frac{tN}{10} - F}{f} \times c\right)
• For the continuous series, the Persontile is P_{t} = l + \left(\frac{\frac{tN}{100} - F}{f} \times c\right)
• For the individual series, the number is which is occured more number of times is considered as mode.
• For the discrete series, the variate which is having maximum frequency is considered mode
• For continuous series, the mode = l + \left(\frac{f_{m} - f_{1}}{2f_{m} - f_{1} - f_{2}} \times c\right)
l = lower limit
fm = Frequency of class
f1 = frequency preceding class
f2 = frequency succeeding class
c = class size
• Mode = 3 Median − 2 Mean
• Mean − Mode = 3(Mean − Median)
• In Symmetric distribution, the upper and lower quartiles are equidistant from median
• For symmetric distribution median = Second Quartile (Q2) = 5th decile (D5) = 50th persontile (P50)
• The difference between the maximum and minimum items of the series is called range
• Coefficient of Range =\tt \frac{Range}{maximum + minimum}
• Quartile deviation = \tt \frac{Q_{3} - Q_{1}}{2}
• Coefficient of quartile deviation = \tt \frac{Q_{3} - Q_{1}}{Q_{3} + Q_{1}}

### View the Topic in this video From 21:30 To 24:30

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