\displaystyle \textbf{Statistics}
\displaystyle \text{The word ``Statistics'' is used in both its singular and plural senses.}
\displaystyle \textbf{1. Singular Sense: }\text{Statistics is the science of collection, presentation,}
\displaystyle \text{analysis and interpretation of numerical data.}
\displaystyle \textbf{2. Plural Sense: }\text{Statistics means numerical facts or observations collected}
\displaystyle \text{for a definite purpose.}
\displaystyle \\
\displaystyle \textbf{Statistical Data}
\displaystyle \textbf{Definition: }\text{Statistical data are of two types: Primary Data and Secondary Data.}
\displaystyle \\
\displaystyle \textbf{1. Primary Data}
\displaystyle \textbf{Definition: }\text{Primary data are collected first-hand by the investigator or researcher.}
\displaystyle \text{They are collected directly from original sources using surveys, interviews,}
\displaystyle \text{experiments or observations for a specific research purpose.}
\displaystyle \\
\displaystyle \textbf{2. Secondary Data}
\displaystyle \textbf{Definition: }\text{Secondary data are data collected by other persons or organizations.}
\displaystyle \text{They are obtained from published reports, surveys, experiments or research studies.}
\displaystyle \text{Researchers generally study secondary data before collecting primary data.}

\displaystyle \textbf{Presentation of Data}
\displaystyle \textbf{Definition: }\text{Presentation of data is the systematic arrangement of data in the form}
\displaystyle \text{of tables, graphs or charts to facilitate analysis and interpretation.}

\displaystyle \textbf{Methods of Presenting Raw Data}
\displaystyle \text{Raw data may be arranged in any one of the following ways:}
\displaystyle \text{1. Serial (or Alphabetical) Order}
\displaystyle \text{2. Ascending Order}
\displaystyle \text{3. Descending Order}

\displaystyle \textbf{Array}
\displaystyle \textbf{Definition: }\text{Raw data arranged in ascending or descending order of magnitude}
\displaystyle \text{is called an array.}

\displaystyle \textbf{Raw (Ungrouped) Data}
\displaystyle \textbf{Definition: }\text{Data collected in its original form without any arrangement is called}
\displaystyle \text{raw data or ungrouped data.}

\displaystyle \textbf{Importance of Arranging Data}
\displaystyle \text{Raw data do not provide sufficient information for analysis or interpretation.}
\displaystyle \text{Arranging data in ascending or descending order gives a clearer picture of the data.}
\displaystyle \text{It helps in understanding the distribution and comparing observations easily.}

\displaystyle \textbf{Ascending Order}
\displaystyle \text{In ascending order, the data look as follows:}
\displaystyle 1,\ 1,\ 5,\ 5,\ 8,\ 10,\ 12,\ 12,\ 12,\ 12,\ 17,\ 17,\ 17,\ 19,\ 19,
\displaystyle 19,\ 19,\ 21,\ 21,\ 21,\ 21,\ 25,\ 33,\ 33,\ 33,\ 39,\ 40,\ 41,\ 41,\ 41

\displaystyle \textbf{Descending Order}
\displaystyle \text{In descending order, the data look as follows:}
\displaystyle 41,\ 41,\ 41,\ 40,\ 39,\ 33,\ 33,\ 33,\ 25,\ 21,\ 21,\ 21,\ 21,\ 19,\ 19,
\displaystyle 19,\ 19,\ 17,\ 17,\ 17,\ 12,\ 12,\ 12,\ 12,\ 10,\ 8,\ 5,\ 5,\ 1,\ 1

\displaystyle \textbf{Array (Arrayed Data)}
\displaystyle \textbf{Definition: }\text{Raw data arranged in ascending or descending order of magnitude}
\displaystyle \text{is called an array or arrayed data.}

\displaystyle \textbf{Need for Frequency Distribution}
\displaystyle \text{When the number of observations is large, arranging data in ascending,}
\displaystyle \text{descending or serial order becomes tedious and less informative.}
\displaystyle \text{Such an arrangement shows only the minimum and maximum values easily.}
\displaystyle \text{Therefore, data are arranged into a frequency distribution table.}

\displaystyle \textbf{Frequency Distribution}
\displaystyle \textbf{Definition: }\text{A table showing each observation together with the number of times}
\displaystyle \text{it occurs is called a frequency distribution.}

\displaystyle \textbf{Frequency Distribution Table}

\displaystyle \begin{array}{|c|c|c|} \hline \textbf{Marks} & \textbf{Tally Marks} & \textbf{Frequency} \\ \hline 1 & || & 2 \\ \hline 5 & || & 2 \\ \hline 8 & | & 1 \\ \hline 10 & | & 1 \\ \hline 12 & |||| & 4 \\ \hline 17 & ||| & 3 \\ \hline 19 & |||| & 4 \\ \hline 21 & |||| & 4 \\ \hline 25 & | & 1 \\ \hline 33 & ||| & 3 \\ \hline 39 & | & 1 \\ \hline 40 & | & 1 \\ \hline 41 & ||| & 3 \\ \hline \end{array}

\displaystyle \textbf{Variate}
\displaystyle \textbf{Definition: }\text{Each observation or value of the variable is called a variate.}
\displaystyle \text{In the above table, the marks are the variates.}

\displaystyle \textbf{Frequency}
\displaystyle \textbf{Definition: }\text{The number of times a particular observation occurs in a data set}
\displaystyle \text{is called its frequency.}
\displaystyle \text{In the above table, the number of students obtaining a particular mark}
\displaystyle \text{is the frequency of that variate.}

\displaystyle \textbf{Grouped Frequency Distribution}
\displaystyle \textbf{Definition: }\text{When observations are grouped into suitable class intervals,}
\displaystyle \text{the resulting table is called a grouped frequency distribution.}

\displaystyle \textbf{Class Interval (or Class)}
\displaystyle \textbf{Definition: }\text{A group of observations lying within a specified range of values}
\displaystyle \text{is called a class or class interval.}

\displaystyle \textbf{Need for Grouping Data}
\displaystyle \text{When the number of observations is large, presenting each value separately}
\displaystyle \text{becomes lengthy and difficult to interpret.}
\displaystyle \text{Grouping observations into class intervals makes the data compact,}
\displaystyle \text{systematic and easier to analyse.}

\displaystyle \textbf{Grouped Frequency Distribution Table}
\displaystyle \begin{array}{|c|c|} \hline \textbf{Class Interval} & \textbf{Frequency} \\ \hline 1\text{--}10 & 6 \\ \hline 11\text{--}20 & 11 \\ \hline 21\text{--}30 & 5 \\ \hline 31\text{--}40 & 4 \\ \hline 41\text{--}50 & 4 \\ \hline \end{array}

\displaystyle \textbf{Frequency Distribution}
\displaystyle \textbf{Definition: }\text{A frequency distribution shows the number of times each observation}
\displaystyle \text{or group of observations occurs in a data set.}
\displaystyle \text{It may be presented in the form of a list, table or graph.}

\displaystyle \textbf{Types of Frequency Distribution}
\displaystyle \text{There are two types of frequency distributions:}
\displaystyle \text{1. Discrete Frequency Distribution}
\displaystyle \text{2. Continuous (or Grouped) Frequency Distribution}

\displaystyle \textbf{Discrete Frequency Distribution}
\displaystyle \textbf{Definition: }\text{A frequency distribution in which each distinct observation is listed}
\displaystyle \text{separately along with its frequency is called a discrete frequency distribution.}
\displaystyle \text{The frequency of an observation is the number of times it occurs in the data set.}

\displaystyle \textbf{Preparation of a Discrete Frequency Distribution}
\displaystyle \text{1. List all the distinct observations.}
\displaystyle \text{2. Count the number of times each observation occurs using tally marks.}
\displaystyle \text{3. Record the count as the frequency of that observation.}

\displaystyle \textbf{Example: Discrete Frequency Distribution}
\displaystyle \text{The following table shows the discrete frequency distribution of the given marks.}

\displaystyle \begin{array}{|c|c|c|} \hline \textbf{Marks} & \textbf{Tally Marks} & \textbf{Frequency} \\ \hline 1 & ||||/ & 5 \\ \hline 2 & ||||/\,| & 6 \\ \hline 3 & |||| & 4 \\ \hline 4 & ||| & 3 \\ \hline 5 & || & 2 \\ \hline \end{array}

\displaystyle \textbf{Observation}
\displaystyle \text{The frequency of each mark is obtained by counting the number of times it appears}
\displaystyle \text{in the data set. Tally marks make counting easier and help prepare the table quickly.}

\displaystyle \textbf{Continuous or Grouped Frequency Distribution}
\displaystyle \textbf{Definition: }\text{A grouped frequency distribution is a table showing class intervals}
\displaystyle \text{together with their corresponding frequencies.}

\displaystyle \textbf{Need for Grouping Data}
\displaystyle \text{When the number of observations is large and the range of the data is wide,}
\displaystyle \text{the observations are grouped into suitable class intervals.}
\displaystyle \text{This makes the data more compact, systematic and easier to analyse.}

\displaystyle \textbf{Example}
\displaystyle \text{The marks obtained by 30 students of Class X in a class test, out of 50 marks,}
\displaystyle \text{can be grouped into suitable class intervals to form a grouped frequency distribution.}

\displaystyle \textbf{Example: Grouped Frequency Distribution}
\displaystyle \text{The grouped frequency distribution of the given marks is shown below.}

\displaystyle \begin{array}{|c|c|c|} \hline \textbf{Class Interval} & \textbf{Tally Marks} & \textbf{Frequency} \\ \hline 0\text{--}10 & ||||/\,| & 6 \\ \hline 11\text{--}20 & ||||/\,||||/\,| & 11 \\ \hline 21\text{--}30 & ||||/ & 5 \\ \hline 31\text{--}40 & |||| & 4 \\ \hline 41\text{--}50 & |||| & 4 \\ \hline \end{array}

\displaystyle \textbf{Exclusive Method of Classification}
\displaystyle \textbf{Definition: }\text{When the upper limit of one class is the lower limit of the next class,}
\displaystyle \text{the classification is called the exclusive method.}
\displaystyle \text{In this method, the upper limit of a class is not included in that class.}

\displaystyle \textbf{Example:}
\displaystyle \text{In the class interval }0\text{--}10,\text{ a student scoring }10\text{ marks is not included}
\displaystyle \text{in this class. The student is included in the next class }10\text{--}20\text{.}

\displaystyle \textbf{Exclusive Method}
\displaystyle \textbf{Definition: }\text{In the exclusive method, the upper limit of a class is not included}
\displaystyle \text{in that class. It is included as the lower limit of the next class.}

\displaystyle \begin{array}{|c|c|} \hline \textbf{Class} & \textbf{Frequency} \\ \hline 0\text{--}10 & 5 \\ \hline 10\text{--}20 & 12 \\ \hline 20\text{--}30 & 5 \\ \hline 30\text{--}40 & 4 \\ \hline 40\text{--}50 & 4 \\ \hline \textbf{Total} & \mathbf{30} \\ \hline \end{array}

\displaystyle \textbf{Inclusive Method}
\displaystyle \textbf{Definition: }\text{In the inclusive method, the upper limit of a class is included}
\displaystyle \text{in that class.}

\displaystyle \begin{array}{|c|c|} \hline \textbf{Class} & \textbf{Frequency} \\ \hline 0\text{--}9 & 5 \\ \hline 10\text{--}19 & 12 \\ \hline 20\text{--}29 & 5 \\ \hline 30\text{--}39 & 4 \\ \hline 40\text{--}49 & 4 \\ \hline \textbf{Total} & \mathbf{30} \\ \hline \end{array}

\displaystyle \textbf{Exclusive and Inclusive Methods}
\displaystyle \begin{array}{|c|c|c|c|} \hline \multicolumn{2}{|c|}{\textbf{Exclusive Method}} & \multicolumn{2}{c|}{\textbf{Inclusive Method}} \\ \hline \textbf{Class} & \textbf{Frequency} & \textbf{Class} & \textbf{Frequency} \\ \hline 0\text{--}10 & 5 & 0\text{--}9 & 5 \\ \hline 10\text{--}20 & 12 & 10\text{--}19 & 12 \\ \hline 20\text{--}30 & 5 & 20\text{--}29 & 5 \\ \hline 30\text{--}40 & 4 & 30\text{--}39 & 4 \\ \hline 40\text{--}50 & 4 & 40\text{--}49 & 4 \\ \hline \textbf{Total} & \mathbf{30} & \textbf{Total} & \mathbf{30} \\ \hline \end{array}

\displaystyle \textbf{Need for Adjustment in Inclusive Method}
\displaystyle \text{In the exclusive method, the class interval extends up to }50\text{, whereas in the}
\displaystyle \text{inclusive method it extends only up to }49\text{.}
\displaystyle \text{Therefore, when the inclusive method is used, an adjustment is required}
\displaystyle \text{to determine the correct class intervals and ensure continuity.}

\displaystyle \textbf{Conversion of Inclusive Method into Exclusive Method}
\displaystyle \textbf{Rule: }\text{If }a\text{--}b\text{ is a class interval in the inclusive method,}
\displaystyle \text{then the corresponding exclusive class interval is}
\displaystyle \left(a-\frac{h}{2}\right)\text{--}\left(b+\frac{h}{2}\right)
\displaystyle \textbf{where }h=\frac{\text{Lower limit of the next class}-\text{Upper limit of the present class}}{2}

\displaystyle \textbf{Example}
\displaystyle h=\frac{10-9}{2}=0.5
\displaystyle \text{Hence, the inclusive class intervals are converted into exclusive class intervals as follows:}

\displaystyle \begin{array}{|c|c|} \hline \textbf{Class Interval} & \textbf{Frequency} \\ \hline -0.5\text{--}9.5 & 5 \\ \hline 9.5\text{--}19.5 & 12 \\ \hline 19.5\text{--}29.5 & 5 \\ \hline 29.5\text{--}39.5 & 4 \\ \hline 39.5\text{--}49.5 & 4 \\ \hline \end{array}

\displaystyle \textbf{Cumulative Frequency Distribution}
\displaystyle \textbf{Definition: }\text{A cumulative frequency is obtained by successively adding the}
\displaystyle \text{frequencies of the classes in a frequency distribution.}
\displaystyle \text{The resulting table is called a cumulative frequency distribution.}

\displaystyle \textbf{Types of Cumulative Frequency}
\displaystyle \text{1. Less Than Cumulative Frequency}
\displaystyle \text{2. Greater Than Cumulative Frequency}

\displaystyle \textbf{Less Than Cumulative Frequency}
\displaystyle \textbf{Definition: }\text{The cumulative frequencies are obtained by adding the class}
\displaystyle \text{frequencies from the first class downwards.}

\displaystyle \begin{array}{|c|c|} \hline \textbf{Class} & \textbf{Cumulative Frequency} \\ \hline \text{Less than }10 & 5 \\ \hline \text{Less than }20 & 17 \\ \hline \text{Less than }30 & 22 \\ \hline \text{Less than }40 & 26 \\ \hline \text{Less than }50 & 30 \\ \hline \end{array}

\displaystyle \textbf{Greater Than Cumulative Frequency}
\displaystyle \textbf{Definition: }\text{The cumulative frequencies are obtained by adding the class}
\displaystyle \text{frequencies from the last class upwards.}

\displaystyle \begin{array}{|c|c|} \hline \textbf{Class} & \textbf{Cumulative Frequency} \\ \hline \text{Greater than }0 & 30 \\ \hline \text{Greater than }10 & 24 \\ \hline \text{Greater than }20 & 13 \\ \hline \text{Greater than }30 & 8 \\ \hline \text{Greater than }40 & 3 \\ \hline \end{array}

\displaystyle \textbf{Steps to Prepare a Discrete Frequency Distribution}
\displaystyle \textbf{Step 1: }\text{Arrange the given raw data in ascending order.}
\displaystyle \textbf{Step 2: }\text{Draw a table with three columns: Variable, Tally Marks and Frequency.}
\displaystyle \textbf{Step 3: }\text{Write all distinct values of the variable in ascending order.}
\displaystyle \textbf{Step 4: }\text{Count the occurrences of each value using tally marks and record}
\displaystyle \text{the corresponding frequency.}

\displaystyle \textbf{Example 1: Discrete Frequency Distribution}
\displaystyle \textbf{Question: }\text{The ages of }25\text{ students of Class X are given below. Prepare a}
\displaystyle \text{frequency distribution.}

\displaystyle \textbf{Solution:}
\displaystyle \textbf{Frequency Distribution of Ages of }25\text{ Students}

\displaystyle \begin{array}{|c|c|c|} \hline \textbf{Age} & \textbf{Tally Marks} & \textbf{Frequency} \\ \hline 14 & |||| & 4 \\ \hline 15 & ||||/\,||| & 8 \\ \hline 16 & ||||/\,||||/ & 10 \\ \hline 17 & ||| & 3 \\ \hline \textbf{Total} & & \mathbf{25} \\ \hline \end{array}

\displaystyle \textbf{Steps to Construct a Grouped Frequency Distribution}
\displaystyle \textbf{Step 1: }\text{Find the maximum and minimum values in the data.}
\displaystyle \textbf{Step 2: }\text{Decide the number of classes (generally between }5\text{ and }15\text{).}
\displaystyle \textbf{Step 3: }\text{Find the range }=\text{Maximum}-\text{Minimum and determine the class size.}
\displaystyle \textbf{Step 4: }\text{Choose class intervals so that both the minimum and maximum values}
\displaystyle \text{are included.}
\displaystyle \textbf{Step 5: }\text{Record each observation in the appropriate class using tally marks.}
\displaystyle \textbf{Step 6: }\text{Count the tally marks to obtain the frequency of each class.}
\displaystyle \textbf{Step 7: }\text{Verify that the sum of all frequencies equals the total number of observations.}

\displaystyle \textbf{Example 2: Grouped Frequency Distribution}
\displaystyle \textbf{Question: }\text{Construct a grouped frequency distribution with class size }10
\displaystyle \text{for the electricity bills of }30\text{ houses.}

\displaystyle \textbf{Solution:}
\displaystyle \textbf{Maximum Bill}=112,\qquad \textbf{Minimum Bill}=14
\displaystyle \textbf{Range}=112-14=98
\displaystyle \textbf{Class Size}=10
\displaystyle \textbf{Number of Classes}=\frac{98}{10}=9.8\approx10
\displaystyle \text{Hence, }10\text{ class intervals are required.}

\displaystyle \begin{array}{|c|c|c|} \hline \textbf{Class Interval} & \textbf{Tally Marks} & \textbf{Frequency} \\ \hline 14\text{--}24 & |||| & 4 \\ \hline 24\text{--}34 & || & 2 \\ \hline 34\text{--}44 & ||| & 3 \\ \hline 44\text{--}54 & ||| & 3 \\ \hline 54\text{--}64 & | & 1 \\ \hline 64\text{--}74 & || & 2 \\ \hline 74\text{--}84 & ||||/ & 5 \\ \hline 84\text{--}94 & ||| & 3 \\ \hline 94\text{--}104 & ||| & 3 \\ \hline 104\text{--}114 & |||| & 4 \\ \hline \textbf{Total} & & \mathbf{30} \\ \hline \end{array}

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