Revision: Statistics Algebra Maths 1 SSC (English Medium) 10th Standard Maharashtra State Board

Statistics is the area of study dealing with the collection, presentation, and analysis of data as well as drawing meaningful conclusions from the data.

A collection of given facts or figures, usually expressed in numerical form.

Each group into which raw data is divided is called a class interval.

The two values that bound a class interval.

• Lower limit: Smallest value of the class

• Upper limit: Largest value of the class

The difference between the highest and lowest observations.

Range = Highest value − Lowest value

• Inclusive form: Both lower and upper limits are included in the class.
  (Example: 1–10, 11–20)

• Exclusive form: Lower limit is included, but upper limit is excluded.
  (Example: 0–10, 10–20)

Frequency:

The number of times a particular observation occurs.

Frequency Distribution:

A tabular arrangement of data showing the frequency of each observation or class.

In a grouped frequency distribution, the modal class is the class interval that has the highest frequency.

\[\text{Adjustment Factor}=\frac{1}{2}\text{(Lower limit of next class - Upper limit of previous class)}\]

\[\text{Frequency density}=\frac{\mathrm{Frequency}}{\text{Class width}}\]

Direct Method:

\[\bar{x}=\frac{\sum f_ix_i}{\sum f_i}\]

where x_i​ = class mark, f_i​ = frequency

Short-cut (Assumed Mean) Method:

\[\bar{x} = A+\frac{\sum f_id_i}{\sum f_i}\]

where d_i = x_i - A
A is the assumed mean

Step-deviation Method:

\[\bar{x}=a+h\frac{\sum f_iu_i}{\sum f_i}\]

where \[u_i=\frac{x_i-a}{h}\]

h is the class width / common factor

If n is odd:
Median =\[\left(\frac{n+1}{2}\right)\]th observation

If n is even:
Median average of  =\[\frac{n}{2}\mathrm{th}\] and \[\left(\frac{n}{2}+1\right)\mathrm{th}\]observations

\[\mathrm{Median}=l+\frac{\left(\frac{n}{2}-cf\right)}{f}\times h\]

• l = lower limit of median class

• n = total frequency

• cf = cumulative frequency of class before median class

• f = frequency of median class

Classes must be continuous before applying the median formula.

\[\mathrm{Mode}=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h\]

l = lower limit of the modal class,
h = size of the class interval (assuming all class sizes to be equal),
f₁ = frequency of the modal class,
f₀ = frequency of the class preceding the modal class,
f₂ = frequency of the class succeeding the modal class.

\[\text{Central angle}=\frac{\text{Value of component}}{\text{Total value}}\times360^\circ\]

• A bar diagram is used for the comparison of quantities.

• A pie diagram shows data in percentage or proportional form.

• A line graph shows change over time.

• A histogram is used for a grouped frequency distribution.

• A frequency polygon represents a frequency distribution graphically.

• A Histogram is a graphical representation of a grouped frequency distribution using rectangles.

• It is used for continuous grouped data.

• Class intervals are shown on the X-axis.

• Frequencies are shown on the Y-axis.

• Rectangles are drawn without gaps between them.

• The height of each rectangle is proportional to its frequency.

• A frequency polygon is a graph obtained by joining the points
  (class-mark, frequency) by straight line segments.

• Class-mark = midpoint of the class interval.

• Two imagined classes (with zero frequency) are taken at the beginning and end to close the polygon.

• A frequency polygon is drawn on the same axes as the histogram (if a histogram is given).

• The polygon starts and ends on the x-axis.

• A pie diagram represents data using a circle.

• The whole circle = total data = 360°.

• Each part of the data is shown by a sector.

• The central angle of a sector is proportional to the data value.

• Larger value → larger sector, smaller value → smaller sector.