## Statistics

# Measures of Dispersion

- If x
_{1}, x_{2}, x_{3}....... x_{n}are the n values of x then its arithmetic mean

\overline{x} = \frac{\sum x_i}{n} - If x
_{1}, x_{2}, x_{3}....... x_{n}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 x
_{1}, x_{2}, ....... x_{n}are the n values of x and its corresponding frequencies are f_{1}, f_{2}, ...... f_{n}, then its arithmetic mean

\tt \overline{x} = \frac{\sum f_i \ x_i}{N}

where N = Σ f_{i} - If x
_{1}, x_{2}, ....... x_{n}are the n values of x and its corresponding frequencies are f_{1}, f_{2}, ...... f_{n}, then its arithmetic mean

\tt \overline{x} = A + \frac{\sum f_i \ d_i}{N}

where d_{i}= x_{i}−A

N = Σ f_{i}

A = Assumed mean - If x
_{1}, x_{2}, ....... x_{n}are the n values of x and its corresponding sizes n_{1}, n_{2}, ...... n_{n}, then its arithmetic mean is given by

\tt \overline{x} = \frac{\sum n_i \ x_i}{\sum n_i} - If x
_{1}, x_{2}, ....... x_{n}are the n values of x and its corresponding weights w_{1}, w_{2}, ...... w_{n}, 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 x
_{1}, x_{2}, ....... x_{n}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 x
_{1}, x_{2}, ....... x_{n}are the mid values of n classes and its corresponding frequencies f_{1}, f_{2}, f_{3}, ..... f_{n}.

Then GM = \sqrt[N]{x_{1}^{f_{1}}, x_{2}^{f_{2}} ...... x_{n}^{f_{n}}}

where N = Σ f_{i} - GM = Anti log \left(\frac{\sum_{i = 1}^{n} f_i \log (x_i)}{N}\right)

where N = Σ f_{i} - If G
_{1}, G_{2}, ....... G_{n}are the GM of the n series of sizes n_{1}, n_{2}, ...... n_{n}, 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 = n_{1}+ n_{2}+ ...... + n_{n} - GM = Antilog \tt \left(\frac{\sum_{i = 1}^{n} n_i \ \log (G_i)}{N}\right)

where N = n_{1}+ n_{2}+ ...... + n_{n} - If x
_{1}, x_{2}, x_{3}....... x_{n}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 x
_{1}, x_{2}, ....... x_{n}are the n values of x and its corresponding frequencies are f_{1}, f_{2}, ...... f_{n}, 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}

d_{1}, d_{2}, ....... d_{n }are the distances, t_{1}, t_{2}, ....... t_{n}are the times, v_{1}, v_{2}, ....... v_{n }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 d
_{1}= d_{2}= ....... = d_{n}. Then the average velocity = \tt \frac{n}{\frac{1}{v_{1}} + \frac{1}{v_{2}} + ..... + \frac{1}{v_{n}}} - If x
_{1}, x_{2}, ....... x_{n}are the n-observations. Then median = \tt \left(\frac{n + 1}{2}\right)^{th} observation, if n is odd - If x
_{1}, x_{2}, ....... x_{n}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 x
_{1}, x_{2}, x_{3}....... x_{n}are the n-values of x and its corresponding frequencies are f_{1}, f_{2}, ....... f_{n}. Then median = \tt \left(\frac{N + 1}{2}\right)^{th} item, if N is odd

where N = Σ f_{i} - If x
_{1}, x_{2}, x_{3}....... x_{n}are the n-values of x and its corresponding frequencies are f_{1}, f_{2}, ....... f_{n}. 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 f_{i} - 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

f_{m}= Frequency of class

f_{1}= frequency preceding class

f_{2}= 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 (Q
_{2}) = 5^{th}decile (D_{5}) = 50^{th}persontile (P_{50}) - 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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