Definitions [6]
Data is a collection of facts, information, or observations, usually in numerical form.
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Example: The heights of all the children in your class.
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Example: The marks obtained by 20 students in a math test.
Raw Data is the information that you have just collected, and it is completely unsorted or unarranged. It's like a messy pile of clothes!
Example (Marks): Imagine a teacher writes the test marks of 15 students on the board:
72, 77, 67, 74, 82, 80, 66, 90, 78, 57, 56, 54, 74, 72, 92
This long list is Raw Data. It's hard to quickly tell who scored the highest or lowest mark.
Frequency Distribution Table: When the number of observations in an experiment is large then we can convert it into the tabular form which is called a Frequency Distribution Table.
Ungrouped Frequency Distribution Table: When the frequency of each class interval is not arranged or organized in any manner.
Grouped Frequency Distribution Table: The frequencies of the corresponding class intervals are organised or arranged in a particular manner, either ascending or descending.
Inclusive or discontinuous Frequency Distribution: A frequency distribution in which the upper limit of one class differs from the lower limit of the succeeding class is called an Inclusive or discontinuous Frequency Distribution.
Exclusive or continuous Frequency Distribution: A frequency distribution in which the upper limit of one class coincides from the lower limit of the succeeding class is called an exclusive or continuous Frequency Distribution.
A circular diagram divided into sectors whose angles are proportional to the corresponding data values is called a pie diagram or pie chart.
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The complete circle represents the total value and measures 360°.
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Each sector represents one component of the total.
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A larger component produces a larger sector.
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Equal components produce equal sectors.
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.
Formulae [1]
\[\text{Central angle}=\frac{\text{Value of component}}{\text{Total value}}\times360^\circ\]
Symbol Form
Let:
\[ x \] = component value
\[ T \] = total value, where \[ T > 0 \]
\[ \theta \] = central angle
Then:
\[ \theta = \frac{x}{T} \times 360^{\circ} \]
Key Points
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A pie diagram represents data using a circle.
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The whole circle = total data = 360°.
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Each part of the data is shown by a sector.
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The central angle of a sector is proportional to the data value.
- Larger value → larger sector, smaller value → smaller sector.
- A Histogram is a graphical representation of a grouped frequency distribution using rectangles.
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It is used for continuous grouped data.
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Class intervals are shown on the X-axis.
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Frequencies are shown on the Y-axis.
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Rectangles are drawn without gaps between them.
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The height of each rectangle is proportional to its frequency.
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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.
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Two imagined classes (with zero frequency) are taken at the beginning and end to close the polygon.
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A frequency polygon is drawn on the same axes as the histogram (if a histogram is given).
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The polygon starts and ends on the x-axis.
