11.3 Ogives (EMBK7)

Cumulative histograms, likewise known together ogives, room graphs that have the right to be used to recognize how many data worths lie above or below a certain value in a data set. The accumulation frequency is calculated native a frequency table, by including each frequency come the complete of the frequencies of every data values prior to it in the data set. The last value for the cumulative frequency will always be equal to the total variety of data values, due to the fact that all frequencies will already have been included to the vault total.

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An ogive is drawn by

plotting the beginning of the very first interval in ~ a (y)-value the zero; plot the end of every interval in ~ the (y)-value same to the cumulative count for the interval; and connecting the clues on the plot with right lines. In this way, the finish of the last interval will always be in ~ the total number of data due to the fact that we will certainly have added up across all intervals.

Worked instance 8: accumulation frequencies and ogives


Determine the cumulative frequencies of the following grouped data and complete the table below. Use the table to attract an ogive of the data.

Interval Frequency Cumulative frequency
(10 (20 (30 (40 (50 Interval Frequency Cumulative frequency
(10 temp text

Ogives carry out look comparable to frequency polygons, i m sorry we observed earlier. The most important difference in between them is that an ogive is a plot of cumulative values, whereas a frequency polygon is a plot that the values themselves. So, to acquire from a frequency polygon come an ogive, we would add up the counts as we move from left to right in the graph.

Ogives are valuable for determining the median, percentiles and five number an introduction of data. Remember that the mean is merely the value in the middle when us order the data. A quartile is merely a 4 minutes 1 of the way from the beginning or the finish of an notified data set. Through an ogive we already know how many data values are over or below a details point, so that is simple to find the middle or a 4 minutes 1 of the data set.

Worked instance 9: Ogives and also the 5 number summary


Use the following ogive come compute the five number review of the data. Remember that the five number summary consists that the minimum, every the quartiles (including the median) and also the maximum.

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Find the quartiles

The quartiles are the worths that room (frac14), (frac12) and (frac34) that the way into the ordered data set. Here the counts go approximately ( ext40), so us can uncover the quartiles by looking in ~ the values corresponding to counts of ( ext10), ( ext20) and also ( ext30). ~ above the ogive a count of

( ext10) corresponds to a worth of ( ext3) (first quartile); ( ext20) corresponds to a worth of ( ext7) (second quartile); and also ( ext30) coincides to a value of ( ext8) (third quartile).

Write down the five number summary

The five number an overview is ((1; 3; 7; 8; 10)). The box-and-whisker plot the this data set is provided below.

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Ogives



Use the ogive come answer the inquiries below.

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Draw the histogram corresponding to this ogive.

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To draw the histogram we require to determine the count in each interval.

Firstly, us can discover the intervals through looking wherein the points space plotted ~ above the ogive. Due to the fact that the points room at (x)-coordinates the (- ext25); (- ext15); (- ext5); ( ext5); ( ext15) and ( ext25), it method that the intervals room (<-25;-15)), etc.

To obtain the counting in every interval us subtract the cumulative counting at the begin of the interval native the cumulative counting at the end of the interval.

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Interval (<-25;-15)) (<-15;-5)) (<-5;5)) (<5;15)) (<15;25))
Count ( ext15) ( ext30) ( ext10) ( ext35) ( ext10)

From this counts us can draw the adhering to histogram:

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The table listed below shows the number of people in each period bracket of broad ( ext8).

Interval (<0;8)) (<8;16)) (<16;24)) (<24;32)) (<32;40))
Count ( ext6) ( ext1) ( ext3) ( ext1) ( ext4)
Cumulative ( ext6) ( ext7) ( ext10) ( ext11) ( ext15)
Interval (<40;48)) (<48;56)) (<56;64)) (<64;72)) (<72;80))
Count ( ext2) ( ext1) ( ext2) ( ext1) ( ext3)
Cumulative ( ext17) ( ext18) ( ext20) ( ext21) ( ext24)

From this table we can attract the accumulation frequency plot: