Formula Sheet – Statistics – Class X

1. The mean \bar x of n value x_1,x_2,x_3.....x_n is given by \bar x=\dfrac{x_1+x_2+x_3+.....x_n}{n}

2. Mean of grouped in date (without class-intervals)

(i) Direct Method : If the frequencies of n observations x_1,x_2,x_3.....x_n\ \text{be}\ f_1,f_2,f_3,.....f_n respectively, then the mean \bar x is given by

\bar x=\dfrac{x_1f_1+x_2f_2+x_3f_3+.....x_nf_n}{f_1+f_2+f_3+.....+f_n}=\dfrac{\sum f_i x_i}{\sum f_i}

(ii) Deviation Method or Assumed Mean Method

In this case, the mean \bar x is given by \bar x=a+\dfrac{\sum f_i\ (x_i-a)}{\sum\ f_i}=a+\dfrac{\sum f_i\ d_i}{\sum f_i} ,

Where, a = assumed mean, \sum f_i = total frequency,  d_i=x_i-a

 \sum f_i\ (x_i-a) = sum of the products of deviations and corresponding frequencies.

3. Mean of grouped data (with class-intervals)

In this case the class marks are treated as x_i

Class mark =\dfrac{Lower\ class\ limit + Upper\ class\ limit}{2}

(i) Direct Method

If the frequencies corresponding to the class marks x_1, x_2, x_3,............ \text{be}\ f_1,f_2,f_3,.....f_n respectively, then mean \bar x is given by \bar x=\dfrac{f_1x_1+f_2x_2+f_3x_3+.....+f_nx_n}{f_1+f_2+f_3+....+\ f_n}=\dfrac{\sum f_i\ x_i}{\sum f_i}

(ii) Deviation or Assumed Mean Method

in this case the mean \bar x is given by \bar x=a=\dfrac{\sum f_id_i}{\sum f_i}

Where, a= assumed mean, \sum f_i= total frequency and  d_i=x_i-a

(iii) Step Deviation Method
In this case we use the following formula.

\bar x\ a+\dfrac{\sum f_i\left(\dfrac{x_i-a}{h}\right)}{\sum f_i}\times a+h\left(\dfrac{\sum f_iu_i}{\sum f_i}\right)

Where, a = assumed mean, \sum f_i= total frequency, h = class = size and u_i=\dfrac{x_i-a}{h}

4. Mode is that value among the observations which occurs most often i.e., the value of the observation having the maximum frequency.

5. If in a data more than one value have the same maximum frequency, then the data is said to be multimodal

6. In a grouped frequency distribution, the class which has the maximum frequency is called the modal class.


7. We use the following formula to find the mode of a grouped frequency distribution.

Mode (M_0)=l+\left(\dfrac{f_1-f_0}{2f_1+f_0-f_2}\right)\times h

l = lower limit of modal class,
h = size of the class-interval,
f1 = frequency of the modal class,
f0 = frequency of the class preceding the modal class,
f2 = frequency of the class succeeding the modal class.

8. Median is the value of the middle most item when the data are arranged in ascending or descending order of magnitude.

9. Median of ungrouped data

(i) If the number of items n in the data is odd, then
Median = value of \left(\dfrac{n+1}{2}\right) th item

(ii) If the total number of items n in the data is even, then
\mathrm{Median =\dfrac{1}{2}\Bigg[value\ of \ \dfrac{n}{2}th\ item+value\ of \ \left(\dfrac{n}{2}+1\right)th\ item}\Bigg]

10. Cumulative frequency of a particular value of the variable (or class) is the sum total of all the frequencies up to that value (or the class).

11. There are two types of cumulative frequency distributions.
(i) cumulative frequency distribution of less than type.
(ii) cumulative frequency distribution of more than type.

12. Median of grouped data with class-intervals

In this case, we first find the half of the total frequencies, i.e., \dfrac{n}{2} . The class in which  \dfrac{n}{2} lies is called the median class and the median lies in this class. “We use the following formula for finding the median.

Median (M_e)=l+\left(\dfrac{\dfrac{n}{2}-cf}{f}\right)\times h ,

Where,
l = lower limit of the median class,
n = number of observations,
cf = cumulative frequency of the class preceding the median class
f = frequency of the median class, h = class size.

13. The three measures mean, mode and median are connected by the following relations. Mode = 3 median – 2 mean.

or median \mathrm{=\dfrac{mode}{3}+\dfrac{2\ mean}{3}\ or\ mean=\dfrac{3\ median}{2}-\dfrac{mode}{2}}

14. The graphical representation of a cumulative frequency distribution is called an ogive or cumulative frequency curve.


15. For less than ogive, we plot the points corresponding to the ordered pairs given by (upper limit, corresponding less than cumulative frequency). After joining these points by a free hand curve, we get an ogive of less than type.


16. For more than ogive, we plot the points corresponding to the ordered pairs given by (lower limit, corresponding more than cumulative frequency). After joining these points by a free hand curve, we get an ogive of more than type.


17. Ogive can be used to estimate the median of data. There are two methods to do so.

First method : Mark a point corresponding to \dfrac{n}{2} , where n is the total frequency, on cumulative frequency axis (y-axis). From this point, draw a line parallel to x-axis to cut the orgin at a point. From this point, draw a line perpendicular to the x–axis to get another point. The abscissa of this point gives median.

Second method : Draw both the ogives (less than ogive and more than ogive) on the same graph paper which cut each other at a point From this point, draw a line perpendicular to the x–axis, to get another point. The abscissa of this point gives median.

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