let's see how to calculate cumulative frequency cumulative frequency is the running sum of frequencies the first cumulative frequency is the same the first frequency which is 1. now we add 1 to the second frequency 2 and get 3. we add 3 to 4 and get 7. we add 7 to 3 and get 10. so the cumulative frequencies are 1 3 7 and 10. notice that cumulative frequency always increases if you see it decrease then you must have done something wrong what does cumulative frequency mean it means one value is less than three three values are less than eight seven values are less than sixteen and ten values are less than eighteen with cumulative frequencies ready we can draw the cumulative frequency graph now we need to connect several points with a smooth curve those points are the lowest boundary zero and the upper boundary cumulative frequency of each class the lowest boundary is 1. so the first point is 1 0. now let's look at the first class the upper boundary is 3 and the cumulative frequency is 1. so the second point is 3 1. the upper boundary of the second class is 8 and the cumulative frequency is 3. so the third point is 8 3. we can construct the remaining 2 points the same way now we have 5 points for 4 classes there is always one more point than the number of classes another way to construct these points is to put all boundaries in the x-coordinate the boundaries are 1 3 8 16 18. so we have 5 points with those numbers as the x coordinates then we put zero in the first points y coordinate and cumulative frequency in the rest so the rest of the y coordinates are 1 3 7 10 now we can connect these 5 points with a smooth curve remember to use a smooth curve not straight lines the y axis is always cumulative frequency c the x-axis is whatever the data represents here it is length what does this graph tell us look at the first point we can see that there's no value less than one and look at the third point eight three if we draw two lines from the point to touch the x and y axis we can see that there are three values less than eight it reflects exactly what the cumulative frequency table tells us we know cumulative frequency always increases so the graph will always rise to if you see it go down you must have done something wrong from the cumulative frequency graph we can find measures of the data set first let's find the median median divides the data set in half we know the total number of values n is 10. therefore we need to find the x value when y is equal to half n 5. we locate 5 on the y axis draw a horizontal line to hit the curve and draw a vertical line to hit the x axis this number is the median it is approximately 13 point 1. similarly to find the lower quartile we locate quarter n 2 point 5 on the y axis and find its corresponding x-coordinate on the curve it is approximately 6 point 6. to find the upper quartile we locate 3 quarters of n 7 point 5 on the y-axis and find its corresponding x-coordinate on the curve it is approximately 16.4 we can also find percentile 90 percentile means 90 of values are smaller than this value 90 percent of 10 is 9. so we need to find the x coordinate on the curve when y is 9. we can see that the 90 percentile is 17 point 3. now we have been finding x for y we can also answer greater than or less than questions by finding y for x for example how many values are greater than 10. we find 10 on the x axis draw a vertical line to hit the curve and a horizontal line to hit the y-axis it is 3.8 which means the cumulative frequency is 3.8 therefore there are three values less than 10. the total number is 10 so 7 values are greater than 10. remember these are all estimates and not accurate values we know that we can find the estimated median l q and u q from the cumulative frequency graph we also know the minimum and maximum value 1 and 18. so we can sketch the box and whisk a plot from these the box and the inside line are lined up with lq uq and the median the whiskers extend from the minimum to lq and the maximum to uq if we know the total number of values n we can also draw a cumulative frequency graph from the box and whisker plot we can read the five values from box and whisker and they are the x coordinates if we know n the y coordinates are zero quarter n half n three quarters of n and n for n equals 10 they are zero two point five five seven point five and ten once we have those fives points we can connect them with a smooth curve to get the cumulative frequency graph therefore we can convert between cumulative frequency graph and box and whisker plot cumulative frequency graph translates frequency table to a visual presentation and shows the distribution of group data in the example we gave the total number of values n as 10 only but group data usually has a large n because this is the purpose of grouping data together we don't want to see the raw data because there's too much of it cumulative frequency graph gives us an easy view of a large amount of data however the curve is an approximation and those measures you read from the graph are also estimates they only give you a rough idea and are not accurate in this class we learned how to draw cumulative frequency graphs cumulative frequency is running sum of frequencies to draw a cumulative frequency graph you connect lowest boundary zero and all upper boundary c f with a smooth curve you can find measures by reading the x coordinate of a point on the cumulative frequency graph with corresponding y coordinate cumulative frequency graph and box and whisker plot can be converted to each other if n is given