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STATISTICS
-- B. Kedhar Guhan ---- X D ---- 34 --
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IntroductionIn this PPT, we would first recap what we had learnt in 9th.-
• Histograms : SLIDE 3
• Frequency polygons: SLIDE 4
• Numerical representatives of ungrouped data: SLIDE 5
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Then we sneak-a-peek on-
Central Tendencies of a Grouped Data: SLIDE 7 Grouped Data : SLIDE 8 Mean of a grouped data: SLIDE 9 Direct Method: SLIDE 11 Assumed mean method: SLIDE 13 Step deviation Method: SLIDE 15 Mode: SLIDE 16 Concept of Cumulative Frequency: SLIDE 17• Median: SLIDE 18• Ogives : SLIDE 20
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Histograms A Histogram displays a range of values of a variable that
have been broken into groups or intervals. Histograms are useful if you are trying to graph a large
set of quantitative data It is easier for us to analyse a data when it is
represented as a histogram, rather than in other forms.
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Frequency Polygon Midpoints of the interval of corresponding
rectangle in a histogram are joined together by straight lines. It gives a polygon
They serve the same purpose as histograms, but are especially helpful in comparing two or more sets of data.
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Numerical representatives of Ungrouped data
1. Arithmetic Mean: (or Average)• Sum of all observation divided
by the Number of observation.• Let x1,x2,x3,x4 ….xn be obs.
( thus there are ‘n’ number of scores)
Then Average = (x1+x2+x3+x4 ….+xn)/n
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Median: When the data is arranged in ascending or descending order, the middle observation is the MEDIAN of the data. If n is even, the median is the average of the n/2nd and (n/2+1/2 )nd observation.
Mode: It is the observation that has the highest frequency.
2
3
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A Grouped Data A grouped data is one which is
represented in a tabular form with the observations (x) arranged in ascending\descending order and respective frequencies( f ) given.
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Mean of a Grouped Data
To obtain the mean, 1. First, multiply value of each
observation(x) to its respective frequency( f ).
2. Add up all the obtained values(fx).3. Divide the obtained sum by the total no.
of observations.
MEAN =
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Lets find the mean of the given data.
Marks obtained
(x)
31 33 35 40
No. of students (f )
2 4 2 2
Lets find the Σfx and Σf.x f fx
Xi 31 Fi 2 FiXi 62
Xii 33 Fii 4 FiiXii 144
Xiii 35 Fiii 2 FiiiXiii 70
Xiv 40 Fiv 2 FivXiv 80
Σf = 2+2+2+4 = 10 Σfx = 62+144+70+80 = 356
So, Mean = ΣfxΣf
= 356 =35.6
10
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Often, we come across sets of data with class intervals, like:
Class Interval
10-25
25-40 40-55 55-70 70-85 85-100
No. os students
2 3 7 6 6 6
To find the mean of such data , we need a class mark(mid-point), which would serve as the representative of the whole class.Class Mark = Upper Limit + Lower Limit 2
**This method of finding mean is known as DIRECT METHOD**
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Lets find the class mark of the first class of the given table.
Class Mark = Upper Limit(25) + Lower Limit(10) 2
= 35 = 17.52
Similarly, we can all the other Class Marks and derive this following table:C.I. No. of
students(f )C.M (x)
fx
10-25 2 17.5 35.0
25-40 3 32.5 97.5
40-55 7 47.5 332.5
55-70 6 32.5 375.0
70-85 6 77.5 465.0
85-100 6 92.5 555.0
Total Σf=30 Σfx=1860.0
Now, mean = Σfx Σf
= 1860 30
= 62
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Assumed Mean Method
Another method of finding MEAN:1. Choose one of the observation as the
“Assumed Mean”. [select that xi which is at the centre of x1, x2,…, xn.
2. Then subtract a from each class mark x to obtain the respective d value (x-a).
3. Find the value of FnDn, where n is a particular class; F is the frequency; and D is the obtained value.
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Mean of the data= mean of the deviations =
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Step-Deviation Method
1. Follow the first two steps as in Assumed Mean method.
2. Calculate u = xi-a
3. Now, mean = x = a+h { }
h
Σ fu Σ f ,
Whereh=size of the CIf=frequency of the modal classa= assumed mean
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Mode of a GROUPED DataThe class with the highest frequency is called the MODAL CLASSC.I. No. of
students(f )C.M (x)
fx
10-25 2 17.5 35.0
25-40 3 32.5 97.5
40-55 7 47.5 332.5
55-70 6 32.5 375.0
70-85 6 77.5 465.0
85-100
6 92.5 555.0
Total Σf=30 Σfx=1860
In this set of data, the class “40-55” is the modal class as it has the highest frequency
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Cumulative Frequency
It’s the ‘running total’ of frequencies. It’s the frequency obtained by adding the
of all the preceding classes. When the class is taken as less than
[the Upper limit of the CI], the cumulative frequencies is said to be the less than type.
When the class is taken as more than [the lower limit of the CI], the cumulative frequencies is said to be the more than type.
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MEDIAN of a GROUPED Data
If n ( no. of classes) is odd, the median is {(n+1)/2}nd class.
If n is even, then the median is the average of n/2nd and (n/2 + 1)th class.
Median for a grouped data is given by
Median = l{ }hn/2 - cf f Where
l= Lower Limit of the class n= no. of observationscf= cumulative frequency of the preceding classf= frequency h= class size
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Relationship between the
Central Tendencies
3 Median = Mode + 2 Mean
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Graphical Representation of Cumulative Frequency Distribution
Cumulative frequency distribution can be graphically represented as a cumulative frequency curve( Ogive )
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More than type Ogive: Mark the LL each class
intervals on the x-axis. Mark their corresponding
cumulative frequency on the y-axis.
Plot the points (L.l. , c.f.) Join all the plotted points
by a free hand smooth curve.
This curve is called Less than type ogive
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Less than type Ogive :
Mark the UL each class intervals on the x-axis.
Mark their corresponding cumulative frequency on the y-axis.
Plot the points (U.l. , c.f.) Join all the plotted points by a free hand
smooth curve. This curve is called Less than type
ogive .
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Median from Ogive
METHOD 1 Locate n/2 on the y-axis. From here, draw a line parallel to x-axis,
cutting an ogive ( less/more than type) at a point.
From this point, drop a perpendicular to x-axis.
The point of intersection of this perpendicular and the x-axis determines the median of the data.
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METHOD 2:
Draw Both the Ogives of the data. From the point of intersection of these
Ogives, draw a perpendicular on the x-axis.
The point of intersection of the perpendicular and the x-axis determines.
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THANK YOU-- B. Kedhar Guhan --
-- X D ---- 34 --