The scientific methods of collection of data, its classification and application to commerce and everyday life is called statistics.
A set of observations is called a called a collection of data. These observations should possess some common characteristics. They are recorded and used for further study. The quantities such as age, height and number of students are called variables.
A continuous variable can assume all values within a certain range, e.g., height, weight and age of persons are examples of continuous variables.
A discrete variable can assume only integral values which can be counted. Number of pupils in a school, persons working in a factory are examples of discrete variables.
A list of some important terms is given below.
(i) ungrouped data
(ii) tabulation of data
(iii) range
(iv) frequency
(v) frequency distribution
(vi) tally
(vii) inclusive type of grouped frequency distribution
(viii) exclusive type of grouped frequency distribution
(ix) lower limit and actual lower limit
(x) upper limit and actual upper limit
(xi) class size or class width
(xii) class mark or class mid-interval.
Introduction to beats:
When two waves of nearly same frequency and having a constant initial phase difference propagate simultaneously along the same straight line in the same direction, they superpose in such a way that the intensity of sound at a point in the medium increases and decreases alternatively. This phenomenon of waxing and waning of sound-intensity is called the beats.
The time-interval between two successive waxing is called the beat-period. The number of waxing and waning in one second is called the beat-frequency.
It is to be noted that the distinct beats are heard only if the difference of frequency of two sounding sources be less than 10, because the effect of sound on ear lasts for nearly 0.1 second.
If two tuning forks of exactly the same frequency are sounded together, no beat is heard; but if a little wax is put on a prong of any one fork, so that its frequency is slightly lowered, then on sounding the two forks simultaneously, distinct beats are heard.
The phenomenon of beats is a result of superposition of two coherent waves of nearly the same frequency; travelling simultaneously in the same direction along the same line.
When the two waves travelling in a medium meet in the same phase, the intensity is maximum and when they meet in opposite phase, the intensity is minimum.
If the frequency of two waves is nearly equal, the phase difference at any point of medium varies with time. Hence at the same point, the waves meet alternately in the same phase and in the opposite phase. Therefore the intensity of sound at that point becomes maximum and minimum alternately. This is the beats-phenomena.
Consider two coherent sources of sound S_{1} and S_{2} of frequencies n_{1} and n_{2} respectively such that n_{1} and n_{2} are nearly equal. The two sources begin to send sound wave in the same direction simultaneously.
Now we choose a particular point of observation, where the sound is heard at all instants continuously. We choose the instant when the crests due to two waves meet.
The amplitude and intensity at this instant will be maximum at this point. At this point the next crest of S_{1} will reach slightly earlier than that of S_{2} and so the waves will not be in the same phase.
Graphical Representation of data:
Statics is the collection, organization, and interpretation of data. The collected data can be easilyanalyzedby using graphs.It is wellsaid that one picture is better than a thousand words. The transformation of data through visual methods like graphs is representation of data Usually comparisons among theindividual items are best shown by means of graphs. The representation then becomeseasier to understand than the actual data.
Different types of data representation that are frequently used:
1.Bar graphs
2.Histograms
3.Frequency polygons
4.Pie charts.
A bar graph is a type of visual representation of data in which usually bars of uniform width are drawn withequal spacing between them on one axis (say, the x-axis), depicting the variable. Thevariable are plotted on the other axis (say, the y-axis) and the heights ofthe bars depend on the values of the variable.
Ex:
A family with a monthly income of Rs 20,000 had planned the followingexpenditures per month under various heads
We draw the bar graph of this data in the following steps. Note that the unitin the second column is thousand rupees. So, ‘4’ against ‘grocery’ means Rs 4000.
1. We represent the Heads (variable) on the horizontal axis, we take equal widthsfor all bars and maintain equal gaps in between. Let one Head be represented byone unit.
2. We represent the expenditure (value) on the vertical axis. Since the maximumexpenditure is Rs 5000, we can choose the scale as 1 unit = Rs 1000.
3. To represent grocery, we draw a rectangular bar with width1 unit and height 4 units.
4. Similarly, other Heads are represented by a gap of unit width
Histogram is a visual form of representation of acontinuous frequency distribution i.e. groupedfrequency distribution with no space between therectangles when plotted. Traditionally, class-intervals aretaken along the horizontal axis while therespective class frequencies are taken along thevertical axis. This is similar to bar graph, but it is used for continuous classintervals
Note:The areas of the rectangles are proportional to the frequencies.
Consider the frequency distribution of the weights of 36 students of a class shown in the table below:
Let us represent the data given above graphically as follows:
(i) We chose an appropriate scale on horizontal axis to represent the weights. We can choose the scale as 1 cm = 5 kg. Also, since the first class interval is starting from 30.5 and not zero, we show it on the graph by marking a kink or a break on the axis.
(ii) We represent the number of students (frequency) on the vertical axis on a suitable scale. Since the maximum frequency is 15, we need to choose the scale to accomodate this maximum frequency.
(iii) We now draw rectangles (or rectangular bars) of width equal to the class-size and lengths according to the frequencies of the corresponding class intervals. For example, the rectangle for the class interval 30.5 - 35.5 will be of width 1 cm and length 4.5 cm.
(iv) In this way, we obtain the graph.
Histogram shown below
Frequency polygon is formed by joining midpoints of the tops the adjoining rectangles in the histogram
For the example above frequency polygon is
Pie diagram is a way of graphical representation of data which can be easily analyzed.It is used to compare the value given to the total value.A pie chartis acircularchartdivided into sectors, showing the proportions so that the length of the arc of each sector is proportional to the value it represents.it is also called as Divided Circle Diagram.
To calculate angles when the data given in values of the individual quantities. Then sum up the all the values ofquantities to get total value and use the below formula.
angle = (("Value given")/("Total value") ) x 360
6. And starting from that line mark the angles for each quantity in an order, and put the titles for the sector.
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