When the frequency of each class is added in succession, the total frequency is called cumulative frequency and is denoted by c.f. The average value of data is called mean. data is grouped in individual, discrete and continuous series. When a given distribution arranged either ascending or in descending order then the value that lies in the middle of the distribution is called median.In the given data the most repeated value or data is called mode.
A collection of facts, such as numbers, words, measurements, observations or even just descriptions of things is known as data.
The data which is collected in initial phase which may not be in an order is known as Raw data. This data can't be used to make a decision.
The raw data which well-arranged properly for further analysis after collecting is known as Arrayed data.
An observation is an individual fact or information in the form of numerical figure.
Frequency
The frequency of a particular data value is the number of times the data value occurs. For example, if four students have a score of 70 in mathematics, and then the score of 70 is said to have a frequency of 3. The frequency of a data value is represented by f.
Frequency table
The tabular representation of given data with their frequencies is known as frequency table.
Cumulative Frequency Table
Cumulative Frequency is used to determine the number of observations that lie above (or below) a particular value in a data set. The is calculated using a Cumulative Frequency distribution table, which can be constructed from stem and leaf plots or directly from the data. Constructing Cumulative Frequency Table: (i) Arrange the given data in increasing order. (ii) Go on adding frequency of each class one after another. For example: 62 63 63 60 55 66 65 66 67 62 68 69 73 63 62 60 59 63 70 65 67 62 66 68 65 64 The cumulative frequency of a particular class is the sum of the frequency of that class and the frequencies of all the classes above it.
Cumulative Frequency Table
Class weight (in kg) (X)
Frequency(f)
Cumulative Frequency (c.f)
55 - 59
1
1
59 - 63
7
1 + 7 = 8
63 - 67
11
8 + 11 = 19
67 - 71
6
19 + 6 = 25
71 - 75
1
25 + 1 = 26
When the frequency of each class is added in succession, the total frequency is called cumulative frequency and is denoted by c.f.
Graphical representation of data
Line graph
A line-graph especially useful in the field of statistic and science. Line- graph are more popular than other graphs because the visual characteristics in them show the trends of data clearly and it is easy to draw. A line-graph is a visual comparison of two variables-shown on x-axes and y-axes.
Pie Chart
A pie chart (or a circle chart) is a circular statistical graph, which is divided into slices to illustrate numerical proportion. In a pie chart, the arc length of each slice (and consequently its central angle and area are proportional to the quantity it represents.
Study of pie chart
1. At first, look at the chart to find the number of sectors on the different title.
2. Add all the quantities of each title.
3. Calculate the value of quantity represented by 10 on the basis of 3600 equivalent to total values (quantity) of the sectors.
4. Find the value of each title (sector) separately according to the angles indicated in the pie chart.
Histogram and Cumulative Frequency Curve
Introduction of the Histogram
A histogram is a graphical representation of the distribution of numerical data. Let us see the frequency table and the histogram given below. 110 employee of a finance company pays the following income tax to the government of their monthly income.
Here, the first bar represents 8 taxpayers paying more than Rs.100 but less than Rs. 150. Similarly, the second bar represents 16 tax payers 16 taxpayers paying more than Rs. 150 but less than Rs. 200. The highest bar represents 24 workers paying more than Rs. 300 but less than Rs. 350. In this way, the above figure obtained by drawing a set of bars having the equal base on the x-axis and the height equivalent to the frequency of the given data or the collected data after representing it in the continuous frequency distribution table with various class interval known as the histogram.
Introduction and Construction of an Ogive (cumulative frequency curve)
The numbers of periods per week assigned to 50 teachers in a boarding school is as follows:
No. of Periods
No. of Teachers
0 - 6 6 - 12 12 - 18 18 - 24 24 - 30 30 - 36
3 10 20 10 5 2
This can be represented as in the following table:
No. of periods
Frequency
Cumulative Frequency
0 - 6
3
3
6 - 12
10
3 + 10 = 13
12 - 18
20
13 = 20 = 33
18 - 24
10
33 + 10 = 43
24 - 30
5
43 + 5 = 48
30 - 36
2
48 + 2 = 50
The above cumulative frequency table can be represented in the 'less than type' simple frequency table as in the following:
Periods
Frequency
Less than 6
3
Less than 12
13
Less than 18
33
Less than 24
43
Less than 30
48
Less than 36
50
Representing this data in graph:
The line so formed is called ogive or cumulative frequency curve.
To draw the ogive of any data, 1 . First of all, find the cumulative frequency of each data or class. 2 . Convert the cumulative frequency table into 'less than type' frequency table. 3 . Plot each data of ' less than type ' frequency table and its frequency on the graph and then join the plotted points freely.
Measure of Central Tendency
Arithmetic Mean
The average value of data is called mean or arithmetic mean. Data is grouped into three categories: 1 . Individual Series 2 . Discrete Series 3 . Continuous Series. **This includes the process of calculating the mean of the individual mean of individual series and discrete series only. **
1 . Individual Series
Data expressed without mentioning frequency is called individual series.
2 . Discrete Series
Data having frequency but no class interval is called discrete series.
Median
When a given data arranged either in ascending or in descending order then the value that lies in the middle of the distribution is called median. If there is N number, then the value at the ( \(\frac{N+1}{2}\) ) place is the median. But if N is even, then the average value of the two terms is the median.
Quartiles
Quartiles are those values of a distribution which divide the given data into four equal parts. A quartile is a type of quantile. The first quartile (Q1) is defined as the middle number between the smallest number and the median of the data set. The second quartile (Q2) is the median of the data. The third quartile (Q3) is the middle value between the median and the highest value of the data set. Quartile divides the whole data into 25%, 50%, and 75%.
For individual and discrete series First Quartile (Q1) = the value of (\(\frac{N+1}{4}\))th item Second Quartile (Q2) = the value of 2 \(\times\) (\(\frac{N+1}{4}\))th item = (\(\frac{N+1}{2}\))th item Third Quartile (Q3) = the value of 3 \(\times\) (\(\frac{N+1}{4}\))th item
Things to remember
Median = [\(\frac{N+1}{2}\)]thposition
First quartile(Q1) = [\(\frac{N+1}{4}\)]th position
Second quartile(Q2) = [\(\frac{N+1}{2}\)]th position
Third quartile(Q3) = 3[\(\frac{N+1}{4}\)]th position
It includes every relationship which established among the people.
There can be more than one community in a society. Community smaller than society.
It is a network of social relationships which cannot see or touched.
common interests and common objectives are not necessary for society.
Videos for Statistics
Statistics - Mean, Median, Mode
Statistics intro: Mean, median, and mode | Data and statistics
Statistics: The average | Descriptive statistics | Probability and Statistics | Khan Academy
Here , total percentage = 100% Now , let 100% = total angle formed at the centre. i.e. 100% = 360 or , 1% = \(\frac{360}{100}\) = 3.6
Now , to find out the central angle formed at centre of each for the blood group are as follows : (i). 40% blood of group A = 40 \(\times\) 3.6 = 144 (ii). 10% blood of group B = 10 \(\times\) 3.6 = 36 (iii). 5% blood of group AB = 5 \(\times\) 3.6 = 18 (iv). 45% blood of group O = 45 \(\times\) 3.6 = 162
Here, total percentage = 100% Here , let 100% = 360 \(\therefore\) 1% \(\frac{360}{100}\) = 3.6 Now , to find angle for each part of the body formed at the centre of pie chart are as follows : (i). 45% of muscles = 45 \(\times\) 3.6 = 162 (ii). 18% of bones = 18 \(\times\) 3.6 = 64.8 (iii). 12% of blood = 12 \(\times\) 3.6 = 43.2 (iv). 20% of fats = 20 \(\times\) 3.6 = 72 (v). 5% of others = 5 \(\times\) 3.6 = 18
Here , total time = 24 hrs wE Covert given data into percentage as follows : School's book = \(\frac{7}{24}\) \(\times\) 100% = \(\frac{175}{6}\)% Sleeping work = \(\frac{8}{24}\) \(\times\) 100% = \(\frac{100}{3}\)% Sport's work = \(\frac{2}{24}\) \(\times\) 100% =\(\frac{25}{3}\)% Home work = \(\frac{3}{24}\) \(\times\) 100% = \(\frac{25}{2}\)% Food = 1 \(\frac{1}{2}\) \(\times\) 100% = \(\frac{25}{4}\)%
Misc. work = 2 \(\frac{1}{2}\) \(\times\) 100% = \(\frac{125}{12}\)% Total angle formed at the centre of each pie chart = 360 Here , let 100% = 360 1% = \(\frac{360}{100}\) = 3.6 Now , we find the angles formed at the centre for each of work as follows : School \(\frac{175}{6}\) =\(\frac{175}{6}\) \(\times\) 3.6 = 105 Sleeping \(\frac{100}{3}\) =\(\frac{100}{3}\) \(\times\) 3.6 = 120 Sports \(\frac{25}{3}\) =\(\frac{25}{3}\) \(\times\) 3.6 = 30 Home work \(\frac{25}{2}\) =\(\frac{25}{2}\) \(\times\) 3.6 = 45 Fooding \(\frac{25}{4}\) = \(\frac{25}{4}\) \(\times\) 3.6 = 22.5 Misc. \(\frac{125}{12}\) = \(\frac{125}{12}\) \(\times\) 3.6 = 37.5
Total percentage of consumed fuel = 100% Angle formed at the centre of pie chart = 360 Here , 100% = 360 Now , we convert fuel percet into degree as follows : Wood (12.5%) = 12.5 \(\times\) 3.6 = 45 Electricity (22.5%) = 22.5 \(\times\) 3.6 = 81 Gas (25%) = 25 \(\times\) 3.6 = 90 Kerosene (40%) = 40 \(\times\) = 144
As the scale on X - axis start from 10 , a break is drawn near the ogive and the graph is drawn using scale beginning from 10 but not the origin itself. In order to draw the histogram , take 1cm = 5 units on X- axis representing the class intervals and 1 cm = 1 unit on Y-axis representing the frequencies of given data , and draw rectangles as shown given below :
Now , convert the above table in cumulative frequency table.
Class
Frequency
Cumulative Frequency
10 - 15 15 - 20 20 - 25 25 - 30 30 - 35
2 4 3 2 3
2 6 9 11 14
Now , to draw the CF curve , convert the above table in less than type CF table by including imagined class interval 5 - 10 woth frequency as O show below :
Class
Cumulative Frequency
Less than `10 Less than 15 Less than 20 Less than 25 Less than 30 Less than 35
0 2 6 9 11 14
In order to draw the cumulative frequency curve of the data from above 'less than' frequency table let us take point 1cm = 5 units on X- axis and 1 cm = 2 unit on y- axis and plot the points and joined smothly.
To draw the histogram from the above table , let us mark the points by taking the scale 2cm = 3 units and 1 cm = 1 unit in Y- axis and draw the rectangle as shown in the graph given below :
Now , converting the above table into cumulative frequency table , we have
Class
Frequency
Cumulative Frequency
12 - 15 15 - 18 18 - 21 21 - 24 24 - 27 27 - 30
2 4 3 6 4 1
2 6 9 15 19 20
Again for drawing the ogive , we convert the above table into 'less than' type cumulative frequency table by including imagined class interval 9 - 12 with frequency 0 as follows :
Class
Cumulative Frequency
Less than 12 Less than 15 Less than 18 Less than 21 Less than 24 Less than 27 Less than 30
0 2 6 9 15 19 20
In order to draw the ogive of the data from above 'less than' frequency table , let us plot the points by taking the scale 1cm = 3 units on X - axis representing less than class interval and 1 cm = 2 units on Y- axis representing the cumulative frequencies and join freely as shown in the given below ,
Votes secured by A = 90 (i). Central angle for votes secured by B = 360 - 120 - 90 = 150 We have given 360 =12000 votes 1 = \(\frac{12000}{360}\) votes 150 = \(\frac{12000}{360}\) \(\times\) 150 votes = 5000 votes.
(ii) . Candidate who secured the highest vote By question , votes secured by A = 90 90 = \(\frac{12000}{360}\) \(\times\) 90 votes = 3000 votes. Similarly , votes secured by C = 120 120 = \(\frac{12000}{360}\) \(\times\) 120 votes = 4000 votes.
