What is accumulation Frequency Curve or the Ogive in Statistics

First we prepare the cumulative frequency table, then the accumulation frequencies space plotted against the upper or lower boundaries of the corresponding class intervals. By involvement the point out the curve so acquired is dubbed a accumulation frequency curve or ogive.There room two species of ogives :

less than ogive : Plot the points with the upper limits of the course as abscissae and the corresponding less 보다 cumulative frequencies as ordinates. The points room joined by free hand smooth curve to provide less 보다 cumulative frequency curve or the much less than Ogive. It is a increasing curve.

You are watching: An ogive is also called a cumulative frequency graph

better than ogive : Plot the points v the lower boundaries of the classes together abscissa and also the corresponding Greater 보다 cumulative frequencies as ordinates. Sign up with the points by a totally free hand smooth curve to get the “More 보다 Ogive”. That is a fall curve.

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When the points obtained are join by straight lines, the picture obtained is dubbed cumulative frequency polygon.Less than ogive method:To build a accumulation frequency polygon and an ogive by less than method, we usage the complying with algorithm.AlgorithmStep 1 : begin with the upper boundaries of class intervals and include class frequencies to achieve the accumulation frequency distribution.Step 2 : note upper class borders along X-axis ~ above a an ideal scale.Step 3 : note cumulative frequencies follow me Y-axis ~ above a suitable scale.Step 4 : Plot the points (xi, fi) wherein xi is the upper limit that a class and also fi is corresponding cumulative frequency.Step 5 : join the points acquired in step 4 through a cost-free hand smooth curve to gain the ogive and to obtain the accumulation frequency polygon sign up with the points acquired in action 4 by line segments.

More than ogive method:To construct a accumulation frequency polygon and an ogive by an ext than method, we usage the complying with algorithm.AlgorithmStep 1 : begin with the lower borders of the course intervals and also from the complete frequencysubtract the frequency of each course to acquire the cumulative frequency distribution.Step 2 : mark the reduced class borders along X-axis on a sutiable scale.Step 3 : Mark the cumulative frequencies along Y-axis ~ above a suitable scale.Step 4 : Plot the points (xi, fi) where xi is the lower limit of a class and also fi is matching cumulative frequency.Step 5 : join the points acquired in action 4 by a cost-free hand smooth curve to obtain the ogive and to obtain the accumulation frequency polygon sign up with these clues by heat segments

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Cumulative Frequency Curve or the Ogive Example problems with Solutions

Example 1: draw a less than ogive for the adhering to frequency distribution :

I.Q.Frequency
60 – 702
70 – 805
80 –9012
90 – 10031
100 – 11039
110 – 12010
120 – 1304

Find the average from the curve.Solution: Let united state prepare following table reflecting the accumulation frequencies much more than the top limit.

Class term (I. Q)Frequency (f)Cumulative frequency
60 – 7022
70 – 8052 + 5 = 7
80 –90122 + 5 + 12 = 19
90 – 100312 + 5 + 12 + 31 = 50
100 – 110392 + 5 + 12 + 31 + 39 = 89
110 – 120102 + 5 + 12 + 31 + 39 + 10 = 99
120 – 13042 + 5 + 12 + 31 + 39 + 10 + 4 = 103

Less 보다 ogive :I.Q. Is handled the x-axis. Number of students are significant on y-axis.Points (70, 2), (80, 7), (90, 19), (100, 50), (110, 89), (120, 99), (130, 103), are plotted ~ above graph document and these points space joined by totally free hand. The curve obtained is less than ogive.

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The value \(\fracN2\) = 51.5 is significant on y-axis and also from this point a heat parallel come x-axis is drawn. This heat meets the curve at a allude P. From P attract a perpendicular PN to meet x-axis at N. N represents the median.Here mean is 100.5.Hence, the typical of offered frequency distribution is 100.5

Example 2: The adhering to table mirrors the everyday sales that 230 footpath sellers that Chandni Chowk.

Sales in Rs.No. Of sellers
0 – 50012
500 – 100018
1000 – 150035
1500 – 200042
2000 – 250050
2500 – 300045
3000 – 350020
3500 – 40008

Locate the average of the over data using only the much less than kind ogive.Solution: To draw ogive, we need to have a accumulation frequency distribution.

Sales in Rs.No. That sellersLess than type cumulative frequency
0 – 5001212
500 – 10001830
1000 – 15003565
1500 – 200042107
2000 – 250050157
2500 – 300045202
3000 – 350020222
3500 – 40008230

Less than ogive : Seles in Rs. Are taken on the y-axis and number of sellers are taken top top x-axis. For drawing less than ogive, clues (500, 12), (1000, 30), (1500, 65), (2000, 107), (2500, 157), (3000, 202), (3500, 222), (4000, 230) are plotted top top graph file and these are joined free hand to acquire the much less than ogive.

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The worth \(\fracN2\) = 115 is significant on y-axis and a heat parallel come x-axis is drawn. This heat meets the curve at a point P. Indigenous P draw a perpendicular PN to meet x-axis at median. Typical = 2000.Hence, the typical of offered frequency circulation is 2000.

Example 3: draw the 2 ogives for the adhering to frequency distribution of the weekly earnings of (less than and much more than) number of workers.

Weekly wagesNumber the workers
0 – 2041
20 – 4051
40 – 6064
60 – 8038
80 – 1007

Hence uncover the worth of median.Solution:

Weekly wagesNumber the workersC.F (less than)C.F (More than)
0 – 204141201
20 – 405192160
40 – 6064156109
60 – 803819445
80 – 10072017

Less than curve : Upper borders of course intervals are marked on the x-axis and less than type cumulative frequencies room taken on y-axis. For illustration less than kind curve, points (20, 41), (40, 92), (60, 156), (80, 194), (100, 201) are plotted top top the graph paper and these are joined by complimentary hand to achieve the much less than ogive.

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Greater 보다 ogiveLower boundaries of course interval are marked on x-axis and greater than kind cumulative frequencies space taken on y-axis. For drawing greater than form curve, point out (0, 201), (20, 160), (40, 109), (60, 45) and also (80, 7) are plotted ~ above the graph file and these are joined by complimentary hand to achieve the higher than type ogive. From the suggest of intersection of these curves a perpendicular heat on x-axis is drawn. The allude at i beg your pardon this line meets x-axis determines the median. Here the median is 42.652.

Example 4: adhering to table offers the accumulation frequency the the age of a group of 199 teachers.Draw the much less than ogive and greater 보다 ogive and find the median.

Age in yearsCum. Frequency
20 – 2521
25 – 3040
30 – 3590
35 – 40130
40 – 45146
45 – 50166
50 – 55176
55 – 60186
60 – 65195
65 – 70199

Solution:

Age in yearsLess than cumulative frequencyFrequencyGreater 보다 type
20 – 252121199
25 – 304019178
30 – 359050159
35 – 4013040109
40 – 451461669
45 – 501662053
50 – 551761033
55 – 601861023
60 – 65195913
65 – 7019944

Find the end the frequencies by subtracting previous frequency native the following frequency to get an easy frequency. Currently we deserve to prepare the better than form frequency. Periods are tackled x-axis and number of teachers on y-axis.Less 보다 ogive :Plot the points (25, 21), (30, 40), (35, 90), (40, 130), (45, 146), (50, 166), (55, 176), (60, 186), (65, 195), (70, 199) on graph paper. Join these points cost-free hand to obtain less than ogive.Greater than ogive : Plot the clues (20, 199), (25, 178), (30, 159), (35, 109), (40, 69), (45, 53), (50, 33), (55, 23), (60, 13), (65, 4) top top graph paper. Join these point out freehand to acquire greater 보다 ogive. Typical is the point of intersection that these two curves.

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Here typical is 37.375.

Example 5: adhering to is the age circulation of a team of students. Attract the cumulative frequency polygon, accumulation frequency curve (less 보다 type) and hence obtain the typical value.

AgeFrequency
5 – 640
6 – 756
7 – 860
8 – 966
9 – 1084
10 – 1196
11 – 1292
12 – 1380
13 – 1464
14 – 1544
15 – 1620
16 – 178

Solution: We first prepare the cumulative frequency table by less then method as given listed below :

AgeFrequencyAge much less thanCumulative frequency
5 – 640640
6 – 756796
7 – 8608156
8 – 9669222
9 – 108410306
10 – 119611402
11 – 129212494
12 – 138013574
13 – 146414638
14 – 154415682
15 – 162016702
16 – 17817710

Other than the given course intervals, we assume a class 4-5 before the very first class interval 5-6 v zero frequency.Now, we mark the upper class boundaries (including the imagined class) follow me X-axis ~ above a an ideal scale and also the cumulative frequencies along Y-axis top top a an ideal scale.Thus, we plot the points (5, 0), (6, 40), (7, 96), (8, 156), (9, 222), (10, 306), (11, 402), (12, 494), (13, 574), (14, 638), (15, 682), (16, 702) and (17, 710).These clues are significant and join by heat segments to acquire the accumulation frequency polygon displayed in Fig.

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In order to acquire the cumulative frequency curve, we draw a smooth curve passing through the points debated above. The graph (fig) shows the total variety of students together 710. The median is the age matching to \(\fracN2\,\, = \,\,\frac7102\) = 355 students. In stimulate to find the median, we an initial located the point corresponding come 355th college student on Y-axis. Allow the point be P. From this allude draw a heat parallel come the X-axis cutting the curve in ~ Q. Indigenous this point Q attract a heat parallel to Y-axis and also meeting X-axis in ~ the suggest M. The x-coordinate of M is 10.5 (See Fig.). Hence, typical is 10.5.
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Example 6: The following observations relate come the height of a group of persons. Attract the two type of accumulation frequency polygons and cumulative frequency curves and also determine the median.

Height in cms140–143143–146146–149149–152152–155155–158158–161
Frequency392631456478
Height in cms161–164164–167167–170170–173173–176176–179179–182
Frequency8596726043206

Solution: Less than an approach : We an initial prepare the accumulation frequency table by much less than an approach as given below :

Height in cmsFrequencyHeight less thanFrequency
140–14331433
143–146914612
146–1492614938
149–1523115269
152–15545155114
155–15864158178
158–16178161256
161–16485164341
164–16796167437
167–17072170509
170–17360173569
173–17643176612
176–17920179632
179–1826182638

Other than the given class intervals, us assume a class interval 137-140 before the first class term 140-143 with zero frequency.Now, we note the top class borders on X-axis and also cumulative frequency follow me Y-axis top top a suitable scale.We plot the clues (140, 0), (143, 3), (146, 12), (149, 38), (152, 69), (155, 114), (158, 178), (161, 256),(164, 341), (167, 437), (170, 509), (173, 569), (176, 612),.(179, 632) and 182, 638).

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These points room joined by heat segments to obtain the accumulation frequency polygon as presented in fig. And by a free hand smooth curve to attain an ogive by much less than technique as shown in fig.

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More than technique : we prepare the accumulation frequency table by an ext than method as given below :Other than the given course intervals, we assume the class interval 182-185 with zero frequency.Now, we mark the lower class limits on X-axis and the cumulative frequencies follow me Y-axis on suitable scales come plot the point out (140, 638), (143, 635), (146, 626), (149, 600), (152, 569), (155, 524), (158, 460), (161, 382), (164, 297), (167, 201), (170, 129), (173, 69), (176, 26) and also (179, 6). By joining this points by heat segments, we attain the an ext than kind frequency polygon as displayed in fig. By joining this points by a free hand curve, we obtain more than type cumulative frequency curve as points by a complimentary hand curve, we obtain more than kind cumulative frequency curve as presented in fig.We find that the two types of accumulation frequency curves crossing at point P. From suggest P perpendicular afternoon is attracted on X-axis. The value of height equivalent to M is 163.2 cm. Hence, mean is 163.2 cm.