B. Com. I (OE) Practical No. 5: Graphical representation of data by using Ogive Curves and Locating Quartile Values.

 

Practical No. 5:  Graphical representation of data by using Ogive Curves and Locating Quartile Values.

Ogive Curve:

If we plot frequencies against the value, we get the frequency curve. If instead of plotting frequencies we plot cumulative frequencies to get an ogive curve.

           In frequency curve we plot the frequencies against the value of the variable but in An ogive curve is obtained by plotting the cumulative frequency against the Class limits. There are two type of cumulative frequencies

i.                   Less than cumulative frequency

ii.                 Greater than cumulative frequency

So we get two type of the ogive curve or cumulative frequency curves as

i.                   Less than ogive or less cumulative frequency curve

ii.                 Greater than ogive or cumulative frequency curve

ii.       Less than ogive or less cumulative frequency curve:

For Less than ogive curve is we first add up the frequencies from top to bottom, (i.e. less than cumulative frequencies) and then plotting the less than cumulative frequencies against the upper limit of the corresponding class.

Note that the less than ogive curve is started from the zero value on y-axis.

ii. Greater than ogive or greater than cumulative frequency curve:

  For Greater than ogive curve is we first add up the frequencies from top bottom to top, (i.e. greater than cumulative frequencies) and then plotting the greater than cumulative frequencies against the lower limit of the corresponding class.

Note that the greater  than ogive curve is end at  the zero value on y-axis.

e.g draw the ogive curve for following data.

Marks	No. of Students
0-10	1
10-20	14
20-30	19
30-40	20
40-50	36
50-60	40
60-70	30
70-80	16
80-90	5
90-100	4

 

here  we find the cumulative frequencies.

Marks	No. of Students	Less than cumulative frequency	Greater than cumulative frequency
0-10	1	1	185
10-20	14	15	184
20-30	19	34	170
30-40	20	54	151
40-50	36	90	131
50-60	40	130	95
60-70	30	160	55
70-80	16	176	25
80-90	5	181	9
90-100	4	185	4

 

*Note that: here in ogive curve we draw a dotted line. *

 

For Determining Median and Quartiles Graphically:

The median is a measure of central tendency, along with the mean, used to describe the typical or central value in a data set. The median is a statistical measure that represents the middle value of a data set when it is arranged in ascending or descending order.

It is the most preferred measure of location for asymmetric distributions and is also called a positional average. The median is defined as the value or observation that lies at the center when all observations are arranged in order of magnitude.

The median divides the data set into two equal parts, with 50% of the data values below the median and 50% above. It is particularly useful when dealing with skewed distributions or data that contains outliers or extreme values, since the median is not affected by such values, unlike the mean.

 

Calculation of Median (Ungrouped Data):

To calculate the median, we first arrange the data in ascending or descending order. The calculation of the median depends on the total number of observations in the data set:

• If the number of observations is odd, the median is the middle value.

Example:
Observations: 14, 20, 17, 18, 15
→ Arranged: 14, 15, 17, 18, 20  (arrange the data in ascending or descending order)
→ Number of observations = 5 (odd) (if total number of observations is odd there is one observation at the center hence the median value is the middle observation  e.g. if the observations are 14, 20, 17, 18, 15. then firstly we arranging in ascending order as 14, 15, 17, 18, 20. here total number of observation is 5 and it is odd then at the middle only one value is median = 17.)
→ Median = 3rd value = 17

• If the number of observations is even, the median is the average of the two middle values.

Example:
Observations: 20, 100, 25, 14, 16, 10
→ Arranged: 10, 14, 16, 20, 25, 100
→ Number of observations = 6 (even)
→ Middle values = 16 and 20

 (i.e. if the total number of observation is  even  then data set contain two value at the middle then we take average of two middle value and consider as median.) e.g. the observations are 20, 100, 25, 14, 16, 10. here total observations are 6 it is even then the arranging in ascending order as 10, 14, 16, 20, 25, 100  in this case at the middle two values 15 and 20 then we take its average as (16+20)/2 =  18)
→ Median = (16 + 20) / 2 = 18

Even if we change the last value (100) to 1000, the median remains the same:

Data: 10, 14, 16, 20, 25, 1000
→ Median = 18

This shows that the median is not affected by extreme values, since it is a positional average. It is widely used in various fields, including statistics, economics, and the social sciences.

 

Median Formula :

 Median = size of [(N+1)/2]th observation

 

Quartiles:

For determining median we divided data in two equal parts.in same way we can divided whole data into four equal parts we use quartiles. There are three quartiles first, second, third quartiles, second quartile is median. It is clear that the data divided in to four equal parts lie at the quarter, half and three quarters in way to arranging data into ascending or descending order.

It can be given as

Q₁ = size of [(N+1)/4]^th observation

Median =Q₂ = size of [(N+1)/2]^th observation

Q₃ = size of [3(N+1)/4]^th observation

The approximate value for continuous distribution the formula is

Q₁ = size of [N/4]^th observation

Median =Q₂ = size of [N/2]^th observation

Q₃ = size of [N/4]^th observation

 

Graphical Determination of Median and Quartiles (Using Ogive):

To determine the median and quartiles graphically, we use an ogive (cumulative frequency curve).

Steps:

1.     First Prepare a Ogive Curve

2.     To find the median or Q2: Median and second Quartile are same.  Q2

o    Calculate N/2 and locate this value on the Y-axis.

o    Draw a Parallel line (at y=N/2) from this point to meet the ogive curve.

o    From the intersection point on the curve, draw a Perpendicular line to the X-axis.

o    The point where the perpendicular line meets the X-axis represents the value of the Median or Q2

3.     Similarly, for Q1:

o    Calculate N/4 and locate this value on the Y-axis

o    Draw a Parallel line (at y=N/4) from this point to meet the ogive curve.

o    From the intersection point on the curve, draw a Perpendicular line to the X-axis.

o    The point where the perpendicular line meets the X-axis represents the value of the First quartile (Q1)

 

4.     For Q3:

o    Calculate 3N/4 and locate this value on the Y-axis

o    Draw a Parallel line (at y=3N/4) from this point to meet the ogive curve.

o    From the intersection point on the curve, draw a Perpendicular line to the X-axis.

o    The point where the perpendicular line meets the X-axis represents the value of the third quartile (Q3)

 

 e.g draw the ogive curve for following data.

 

Marks	No. of Students
0-10	1
10-20	14
20-30	19
30-40	20
40-50	36
50-60	40
60-70	30
70-80	16
80-90	5
90-100	4

 

here  we find the cumulative frequencies.

Marks	No. of Students	Less than cumulative frequency	Greater than cumulative frequency
0-10	1	1	185
10-20	14	15	184
20-30	19	34	170
30-40	20	54	151
40-50	36	90	131
50-60	40	130	95
60-70	30	160	55
70-80	16	176	25
80-90	5	181	9
90-100	4	185	4

The Quartile Value are determined by graphically

1.     To find the median or Q2: Median and second Quartile are same.  Q2

o    Calculate N/2 and locate this value on the Y-axis. OR (The two ogive curves are intercept at point. draw a  perpendicular from this point to x-axis we get  medina of  given data. therefore we draw both ogive curves to determine the median.) here any one method used to determine the median using ogive curve.

o    Draw a Parallel line (at y=N/2  at 92.5) from this point to meet the ogive curve.

o    From the intersection point on the curve, draw a Perpendicular line to the X-axis.

o    The point where the perpendicular line meets the X-axis represents the value of the Median or Q2 = 50.1

2.     Similarly, for Q1:

o    Calculate N/4 and locate this value on the Y-axis

o    Draw a Parallel line (at y=N/4 = 46.25) from this point to meet the ogive curve.

o    From the intersection point on the curve, draw a Perpendicular line to the X-axis.

o    The point where the perpendicular line meets the X-axis represents the value of the First quartile (Q1) = 36

 

3.     For Q3:

o    Calculate 3N/4 and locate this value on the Y-axis

o    Draw a Parallel line (at y=3N/4 = 138.75) from this point to meet the ogive curve.

o    From the intersection point on the curve, draw a Perpendicular line to the X-axis.

o    The point where the perpendicular line meets the X-axis represents the value of the third quartile (Q3) = 62

Therefore Q1 = 36, Q2 = 50.1 and Q3 = 62

Example for Practice. 

B. Com. I(OE): Basic Statistics Practical - I

Expt. No. 5                                                                                       Date:  /    / 2025

Title: Graphical representation of data by using Ogive Curves and Locating Quartile Values.

Q.1 Draw less than ogive from the following data.

Age in Year	20-24	25-29	30-34	35-39	40-44	45-49	50-54	55-59
No. of Workers	7	12	17	25	65	40	27	10

Q. 2 Draw an Ogive curve corresponding to the following data and compute quartiles.

Wages in Rs	25-30	30-35	35-40	40-45	45-50	50-55	55-60	60-65	65-70
No. of workers	10	13	20	22	24	27	25	11	9

Q.3 Draw two Ogive curve and find median.

Marks	0-5	5-10	10-15	15-20	20-25	25-30	30-35	35-40
No. of Students	2	10	18	25	37	21	14	7

Q.4 Draw an Ogive curve corresponding to the following data and compute quartiles.

Salary in Rs	0-100	100-200	200-300	300-400	400-500
No. of Persons	130	120	105	110	100

Q. 5 Draw greater than ogive curve from the following data.

Class	0-9	10-19	20-29	30-39	40-49	50-59	60-69	70-79
Frequency	2	7	14	30	40	36	18	10