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

The median is the middle value of an arranged data set. It divides the data into two equal parts: 50% of observations lie below it and 50% lie above it.

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

A histogram is a graphical representation of a continuous grouped frequency distribution using adjacent rectangles. Class intervals are marked on the horizontal axis, while frequency or frequency density is marked on the vertical axis.

A graph formed by plotting class marks against their corresponding frequencies and joining consecutive points with straight line segments is called a frequency polygon.

A circular diagram divided into sectors whose angles are proportional to the corresponding data values is called a pie diagram or pie chart.

• The complete circle represents the total value and measures 360°.

• Each sector represents one component of the total.

• A larger component produces a larger sector.

• Equal components produce equal sectors.

\[ \begin{array}{c} \text{The mean of statistical data} = \dfrac{\text{The sum of all scores}}{\text{Total no. of scores}} = \dfrac{\sum\limits_{i = 1}^{N} x_{i}}{N} \end{array} \]

(Here \[ x_{i} \] is the \[ i^{th} \] score)

Mean is denoted by \[ \overline{X} \] and it represents the average of the given data.

\[ \overline{X} = \frac{\sum\limits_{i = 1}^{N} x_{i}}{N} \]

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

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

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

• 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.

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

Symbol Form

Let:

\[ x \] = component value

\[ T \] = total value, where \[ T > 0 \]

\[ \theta \] = central angle

Then:

\[ \theta = \frac{x}{T} \times 360^{\circ} \]

• Statistics deals with collecting, organising, analysing, and interpreting data.

• Central tendency gives a single representative value for a data set.

• Mean is the arithmetic average.

• Median is the middle value of ordered data.

• Mode is the most frequently occurring value.

• Grouped data uses class intervals.
• Class mark \[ x_{i} = \frac{\text{lower limit} + \text{upper limit}}{2}. \]
• Direct method: \[ \bar{x} = \frac{\sum f_{i}x_{i}}{N}. \]
• Assumed mean method: \[ \bar{x} = A + \frac{\sum f_{i}d_{i}}{N}. \]
• Step-deviation method: \[ \bar{x} = A + \left(\frac{\sum f_{i}u_{i}}{N}\right)g. \]
• \[ d_{i} = x_{i} - A, \; u_{i} = \frac{d_{i}}{g}, \; N = \sum f_{i}. \]
• The mean of grouped data is an estimate.
• Always show the calculation table and mention units in board answers.

• The median divides arranged data into two equal parts.
• For ungrouped data:
  Odd \[ n: \] median is the \[ \left(\frac{n + 1}{2}\right) \text{th value.} \]
  Even \[ n: \] median is the average of the \[ \frac{n}{2} \text{th} \] and \[ \left(\frac{n}{2} + 1\right) \text{th values.} \]
• For grouped data, first prepare a cumulative-frequency table.
• The median class is the class where cumulative frequency first becomes greater than or equal to \[ n/2. \]
• Convert inclusive classes into continuous classes before applying the formula.
• Use: \[ \boxed{\text{Median} = l + \frac{\left(\frac{n}{2} - cf\right)}{f} \times h} \]
• The median is preferred when data contains extreme values.

Mode = most frequently occurring value.

For grouped data, first find the modal class.

Modal class = class interval with the highest frequency.

Formula:

\[ \text{Mode} = l + \left(\frac{f_{1} - f_{0}}{2f_{1} - f_{0} - f_{2}}\right)h \]

• A table organises exact data; a graph makes patterns easier to notice.

• 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.