Statistics

STATISTICS

-- B. Kedhar Guhan ---- X D ---- 34 --

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

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

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.

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.

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

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

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.

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 =

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

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**

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

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.

Mean of the data= mean of the deviations =

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

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

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.

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

Relationship between the

Central Tendencies

3 Median = Mode + 2 Mean

Graphical Representation of Cumulative Frequency Distribution

Cumulative frequency distribution can be graphically represented as a cumulative frequency curve( Ogive )

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

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 .

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.

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.

THANK YOU-- B. Kedhar Guhan --

-- X D ---- 34 --