1. INTRODUCTION
In various fields, we need information in the form of numerical figures, called data. The data may be related to various aspects such as the marks obtained by the students of a class in an examination, the monthly wages earned by workers in a factory, the profits of a company during the last few years, etc….
2. BASIC DEFINITIONS
Data
A collection of information in the form of numerical figures, is called data.
Eg: The following table gives the data regarding the favorite sports of 140 students of a school.
Sports |
Number of students |
Cricket |
40 |
Football |
30 |
Tennis |
25 |
Badminton |
20 |
Volleyball |
25 |
Statistics
It is the science which deals with the collection, presentation, analysis and interpretation of numerical data.
Observation
Each numerical figure in a data is called a observation.
Raw Data
When some information is collected and presented randomly then it is called a raw data.
Eg: The marks obtained by 10 students in a test are: 40, 32, 15, 36, 9, 23, 48, 25, 13, 27
Array
An arrangement of numerical figures of a data in ascending or descending order is called an array. Thus the above data may be arranged in an ascending order as: 9, 13, 15, 23, 25, 27, 32, 36, 40, 48
Range
The difference between the highest and the lowest values of observation in a data is called the range of the data. In the above data we have:
Lowest marks obtained = 9, Highest marks obtained = 48
Therefore range of the above data = (48 – 9) = 39
3. MEAN
The arithmetic mean or simply the mean of some given observations is defined as
$\text{Mean}=\frac{\text{Sum of given observations}}{\text{Number of observations}}$
Thus, of ‘n’ observations ${\mathrm{x}}_{1},{\mathrm{x}}_{2},\dots \dots \dots \dots \dots .,{\mathrm{x}}_{\mathrm{n}}$ are given then their mean is given by :
$\text{Mean}=\frac{{x}_{1}+{x}_{2}+\dots \dots +{x}_{n}}{n}=\frac{\sum {x}_{i}}{n}$
Here $\sum $is called sigma. This is a Greek letter showing summation.
Eg: Given are the heights (in cm) of 11 boys of a class: 149, 142, 154, 146, 143, 152, 148, 132, 140, 139, 128.
Arrange the above data in an ascending order and find:
(i) The height of the tallest boy
(ii) The height of the shortest boy
(iii) The range of the given data
(iv) The mean height
Sol: Arranging the given data in an ascending order, we get the heights (in cm) as:
128, 132, 139, 140, 142, 143, 146, 148, 149, 152, 154
from the above data, we have:
(i) The height of the tallest boy = 154cm
(ii) The height of the shortest boy = 128cm
(iii) The range of the given data = (154 – 128)cm = 26cm
(iv) Sum of the given observations = (128 + 132 + 139 + 140+ 142 + 143 +1463 + 148 + 149 + 152 + 154)cm = 1573cm
Therefore
$\text{Mean Height}=\frac{\text{Sum of given observations}}{\text{Number of observations}}=\frac{1573}{11}=143$
Hence, the mean height is 143cm.
4. MEDIAN
Median refers to the value that lies in the middle of the data with half of the observations above it and the other half of the observations below it. The following are the steps to calculate median.
Step – 1: Arrange the data in ascending order.
Step – 2: The value that lies in the middle such that half of the observations lie above it and the other half below it will be the median.
The mean, mode and median are representative values of a group of observations or data, and lie between the minimum and maximum values of the data. They are also called measures of the central tendency.
5. MODE
Mode refers to the observation that occurs most often in a given data. The following are the steps to calculate mode:
Step – 1: Arrange the data in ascending order.
Step – 2: Tabulate the data in a frequency distribution table.
Step – 3: The most frequently occurring observation will be the mode.
6. FREQUENCY DISTRIBUTION
Tabulation of Data
Arranging the data in a systematic form in the form of a table is called tabulation of data.
Frequency of an Observation
The number times of a particular observation occurs is called its frequency.
Mean of Tabulated Data
Let the frequencies of ‘n’ observations ${x}_{1},{x}_{2},{x}_{3},\dots \dots \dots .{x}_{n}\text{be}{f}_{1},{f}_{2},{f}_{3},\dots \dots {f}_{n}$respectively then, we define:
$\text{Mean}=\frac{{\mathrm{f}}_{1}{\mathrm{x}}_{1}+{\mathrm{f}}_{2}{\mathrm{x}}_{2}+{\mathrm{f}}_{3}{\mathrm{x}}_{3}+\dots .+{\mathrm{f}}_{\mathrm{n}}{\mathrm{x}}_{\mathrm{n}}}{{\mathrm{f}}_{1}+{\mathrm{f}}_{2}+{\mathrm{f}}_{3}+\dots .+{\mathrm{f}}_{\mathrm{n}}}=\frac{{\mathrm{f}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}}{\sum {\mathrm{f}}_{\mathrm{i}}}$
7. GRAPHICAL REPRESENTATION OF STATISTICAL DATA
The tabular representation of data calls for close observation and cumbersome calculations to compare and study the given data. However, graphical representation of the same provides quick and interesting study of the given data and a curious glance can reveal several inferences. Thus, with the help of pictures or graphs, data can be compared easily. Though there are various types of graphs, in this chapter, we shall be dealing with the following:
• Pie Chart or Pie Graph
• Bar Graph or Column Graph
• Line Graph
Pie Chart
In a pie-chart, various observations or components are represented by the sectors of a circle and the whole circle represents the sum of the values of all the components. Clearly, the total angle of ${360}^{0}$ at the center of the circle is divided according to the values of the components. Thus we have,
Central Angle for Component $=\left(\frac{\text{Value of the Component}}{\text{Total Value}}\times 360\right)$
Some times, the value of components are expressed in percentages. In such cases, we have:
Central Angle for Component$=\left(\frac{\text{Percentage Value of the Component}}{100}\times 360\right)$
Steps of Construction of a Pie Chart for a given Data
Step1: Calculate the central angle for each component, using the above formula.
Step2: Draw a circle of convenient radius.
Step3: Within the circle, draw a horizontal radius.
Step4: Starting with the horizontal radius, draw radii making central angles corresponding to the values of the respective components, till all the components are exhausted.
Step5: Shade each sector differently and mark the component it represents. Thus, we obtain the required pie-chart for the given data
Bar Graph or Column Graph
Is a pictorial representation of numerical data in the form of rectangles (or bars) of equal width and varying heights. These rectangles are drawn at equal intervals. The height of a rectangular bar (or column) represents the frequency of the corresponding observation.
Steps of Construction of a Bar Graph for a Given Data
Step1: On a graph paper, draw a horizontal line OX (called x-axis) and a vertical line OY (called y-axis).
Step2: Mark point at equal intervals along the x-axis. Below these points, write the names of the data items whose values are to be plotted.
Step3: Choose a suitable scale. On that scale determine the heights of the bars for the given numerical values.
Step4: Mark off these heights parallel to the y-axis from the points taken in step2.
Step5: On the x-axis, draw bars of equal width for the heights marked in step4. These bars represent the given numerical data.
Line Graph
In a line graph, points are plotted on the graph paper related to two variables. These points are joined in pairs by lines to obtain a line graph.
Steps of Construction of a Line Graph for a Given Data
Step1: On a graph paper, draw a horizontal line OX (called x-axis) and a vertical line OY (called y-axis).
Step2: Mark points at equal intervals along the x-axis. Below these points write the names of the data items whose values are to be plotted.
Step3: Choose an appropriate scale along the y-axis taking into consideration the given values.
Step4: Mark off points parallel to y-axis from the point taken into step2.
Step5: Join each point so obtained with the successive point with a straight line, using a ruler. Thus, we obtain the required line graph.
8. CHANCE AND PROBABILITY
It is remarkable that a science which began with the consideration of games of chance should be
elevated to the rank of the most important subject of human knowledge.
In everyday life, we come across statements such as
(1) It will probably rain today.
(2) I doubt that he will pass the test.
(3) Most probably, Kavita will stand first in the annual examination.
(4) Chances are high that the prices of diesel will go up.
(5) There is a 50-50 chance of India winning the toss in today’s match.
The words ‘probably’, ‘doubt’, ‘most probably’, ‘chances’, etc., used in the statements above involve an element of uncertainty.
Experiment
An operation which results in some well defined outcomes is called an Experiment.
Random experiment
Any action which gives one or more results of an experiment is called an outcome of the experiment.
In other words, if an experiment is performed many times under similar conditions and the outcome is not the same each time, then such an experiment is called a random experiment.
Example
Testing of a fair coin is a random experiment because if we toss a coin either heads or tails will come up; but if we toss a coin again and again, the outcome each time will not be the same.
Sample space
The set of all possible outcomes of a random experiment is called the sample space for the experiment.
It is usually denoted by S.
Example
When a coin is tossed, either heads or tails will come up. If ‘H’ denotes the occurrence of heads ‘T’ denotes the occurrence of tails, then
Sample space S = {H, T}
Sample point (or) Event point
Each element of the sample space is called a sample point of an event point.
Example
When a die is thrown, sample space
S = {1, 2, 3, 4, 5, 6}
Here 1, 2, 3, 4, 5 and 6 are the sample points.
Probability
If ‘S’ is the finite sample space of an experiment and every outcome of ‘S’ is equally likely and ‘E’ is an event (i.e., E Ì S), then the probability that ‘E’ takes place is defined as
$P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{\text{number of elements in}E}{\text{number of elements in}S}$