Plotting Frequency Distributions

Mike Slater

Here is some data – the number of pupils in a class with birthdays in each month

Month No of pupils born that month

January 2February 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1

Let’s plot a histogram for this dataMonth No of pupils born that

No of pupils with birthday in a given month

2 pupils had birthdays in January

2 pupils had birthdays in February

1 pupils had a birthdays in March

This is a frequency density plot

Now there is another way of looking at our data

Lets look at it cummatively

Cumulative

January 2 2+ 0 = 2February 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1

January is the first month, so there is no data prior to

Cumulative

January 2 2February 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1

Cumulative

For the other months add the data from the

month in question

Cumulative

For the other months add the data from the

month in question

Cumulative

To the cumulative figure from the previous month

Cumulative

To the cumulative figure from the previous month

Cumulative

January 2 2February 2 2 + 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 2 + 2 = 4March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1

Now do the same for March

Cumulative

January 2 2February 2 4March 1 1 + 4April 4May 5June 5July 4August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1 1 + 4 = 5April 4May 5June 5July 4August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1 5April 4May 5June 5July 4August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1 5April 4May 5June 5July 4August 3September 2October 2November 1December 1

Now do the same for

Cumulative

January 2 2February 2 4March 1 5April 4 4 + 5May 5June 5July 4August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1 5April 4 4 + 5 = 9May 5June 5July 4August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1 5April 4 9May 5June 5July 4August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1 5April 4 9May 5June 5July 4August 3September 2October 2November 1December 1

Now let’s do the same for the

remaining months

Cumulative

January 2 2February 2 4March 1 5April 4 9May 5 14June 5July 4August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4 23August 3September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4 23August 3 26September 2October 2November 1December 1

Cumulative

January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4 23August 3 26September 2 28October 2November 1December 1

Cumulative

January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4 23August 3 26September 2 28October 2 30November 1December 1

Cumulative

January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4 23August 3 26September 2 28October 2 30November 1 31December 1

Cumulative

January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4 23August 3 26September 2 28October 2 30November 1 31December 1 32

Now let’s plot a chart for the cumulative dataMonth No of pupils

born that monthCumulative

January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4 23August 3 26September 2 28October 2 30November 1 31December 1 32

In January 2 pupils had birthdays

In the period January to February 2 + 2 = 4 pupils had birthdays(2 in January and 2 in February)

In the period January to March 5pupils had birthdays(2 in January + 2 in February + 1 in March)

In the period January to April 9 pupils had birthdays(2 in January + 2 in February + 1 in March + 4 in April)

Now let’s do the same for the remaining months

We now have a cumulative frequency plot

Let’s try to apply this to toxicology testing

Dose (mg/kg) No of deaths No of test animals

% deaths

5 0 20 0

10 0 20 0

25 3 20 15

50 7 20 35

100 15 20 75

250 19 20 95

500 20 20 100

1000 20 20 100

% deaths

5 0 20 0

10 0 20 0

25 3 20 15

50 7 20 35

100 15 20 75

250 19 20 95

500 20 20 100

1000 20 20 100

At 5 mg/kg no animals

% deaths

5 0 20 0

10 0 20 0

25 3 20 15

50 7 20 35

100 15 20 75

250 19 20 95

500 20 20 100

1000 20 20 100

At 10 mg/kg no animals

% deaths

5 0 20 0

10 0 20 0

25 3 20 15

50 7 20 35

100 15 20 75

250 19 20 95

500 20 20 100

1000 20 20 100

At 25 mg/kg

3 animals died

% deaths

5 0 20 0

10 0 20 0

25 3 20 15

50 7 20 35

100 15 20 75

250 19 20 95

500 20 20 100

1000 20 20 100

At 50 mg/kg

7 animals died

% deaths

5 0 20 0

10 0 20 0

25 3 20 15

50 7 20 35

100 15 20 75

250 19 20 95

500 20 20 100

1000 20 20 100

But some of these

would have died at a

lower dose

% deaths

5 0 20 0

10 0 20 0

25 3 20 15

50 7 20 35

100 15 20 75

250 19 20 95

500 20 20 100

1000 20 20 100

3 animals died at

25 mg/kg

% deaths

5 0 20 0

10 0 20 0

25 3 20 15

50 7 20 35

100 15 20 75

250 19 20 95

500 20 20 100

1000 20 20 100

So an EXTRA 4

died at the higher dose

% deaths

5 0 20 0

10 0 20 0

25 3 20 15

50 7 20 35

100 15 20 75

250 19 20 95

500 20 20 100

1000 20 20 100

15 animals died at 100

% deaths

5 0 20 0

10 0 20 0

25 3 20 15

50 7 20 35

100 15 20 75

250 19 20 95

500 20 20 100

1000 20 20 100

So an EXTRA 8 died than at lower

The deaths at each administered dose represent the TOTAL number of deaths caused by administering doses UP TO that dose

So we have CUMMULATIVE data

If we plot this data we have a CUMMULATIVE frequency plot

Series10

Administered dose mg/kg body weight

5 10 250

5 10 25 500

5 10 25 50 1000

5 10 25 50 100 2500

5 10 25 50 100 250 5000

5 10 25 50 100 250 500 10000

0 200 400 600 800 1000 12000

Plot of the % deaths against administered dose (as an X Y Plot)

1 10 100 10000

Plotted on a logarithmic scale

1 10 100 10000

We can now determine the LD50

1 10 100 10000

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Plotting Frequency Distributions

Health & Medicine

Transcript of Plotting Frequency Distributions

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