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

Let’s plot a histogram for this data
Month No of pupils born that month
January 2
February 2
March 1
April 4
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

No of pupils with birthday in a given month
1
2 pupils had birthdays in January
0

No of pupils with birthday in a given month
6
5
2 pupils had birthdays in January
4
3
2
1
0

No of pupils with birthday in a given month
6
5
2 pupils had birthdays in February
4
3
2
1
0

No of pupils with birthday in a given month
6
5
1 pupils had a birthdays in March
4
3
2
1
0

No of pupils with birthday in a given month
6
5
4 pupils had birthdays in January
4
3
2
1
0

No of pupils with birthday in a given month
6
5
4
3
2
1
0

No of pupils with birthday in a given month
6
5
4
3
2
1
0

No of pupils with birthday in a given month
6
5
4
3
2
1
0

No of pupils with birthday in a given month
6
5
4
3
2
1
0

No of pupils with birthday in a given month
6
5
4
3
2
1
0

No of pupils with birthday in a given month
6
5
4
3
2
1
0

No of pupils with birthday in a given month
6
5
4
3
2
1
0

No of pupils with birthday in a given month
6
5
4
3
2
1
0

This is a frequency density plot
6
5
4
3
2
1
0

Now there is another way of looking at our data

Now there is another way of looking at our data
Lets look at it cummatively

Month No of pupils born Cumulative
that month
January 2
February 2
March 1
April 4
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2
February 2
March 1
April 4
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2+ 0 = 2
February 2
January is the first
March 1 month, so there is no
April 4 data prior to this
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2
March 1
April 4
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2
March 1
April 4
For the other months
May 5
add the data from the
June 5 month in question
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2
March 1
April 4
For the other months
May 5
add the data from the
June 5 month in question
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2
March 1
April 4
To the cumulative
May 5
figure from the
June 5 previous month
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2
March 1
April 4
To the cumulative
May 5
figure from the
June 5 previous month
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 2+2
March 1
April 4
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 2+2=4
March 1
April 4
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1
April 4
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1
April 4
May 5
June 5
Now do the same
July 4 for March
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 1+4
April 4
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 1+4=5
April 4
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4
May 5
June 5
July 4 Now do the
same for April
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 4+5
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 4+5=9
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 9
May 5
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 9
May 5
June 5
July 4
Now let’s do the same
August 3 for the remaining
September 2 months
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 9
May 5 14
June 5
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 9
May 5 14
June 5 19
July 4
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 9
May 5 14
June 5 19
July 4 23
August 3
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 9
May 5 14
June 5 19
July 4 23
August 3 26
September 2
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 9
May 5 14
June 5 19
July 4 23
August 3 26
September 2 28
October 2
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 9
May 5 14
June 5 19
July 4 23
August 3 26
September 2 28
October 2 30
November 1
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 9
May 5 14
June 5 19
July 4 23
August 3 26
September 2 28
October 2 30
November 1 31
December 1

Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 9
May 5 14
June 5 19
July 4 23
August 3 26
September 2 28
October 2 30
November 1 31
December 1 32

Now let’s plot a chart for the cumulative data
Month No of pupils born Cumulative
that month
January 2 2
February 2 4
March 1 5
April 4 9
May 5 14
June 5 19
July 4 23
August 3 26
September 2 28
October 2 30
November 1 31
December 1 32

35
30
25
20
15
In January 2 pupils had birthdays
10
5
0

35
30
25
20
In the period January to February 2 + 2 = 4 pupils had birthdays
15 (2 in January and 2 in February)
10
5
0

35
30
25
In the period January to March 5pupils had birthdays
20 (2 in January + 2 in February + 1 in March)
15
10
5
0

35
30
In the period January to April 9 pupils had birthdays
25 (2 in January + 2 in February + 1 in March + 4 in April)
20
15
10
5
0

Now let’s do the same for the remaining months
35
30
25
20
15
10
5
0

35
30
25
20
15
10
5
0

35
30
25
20
15
10
5
0

35
30
25
20
15
10
5
0

35
30
25
20
15
10
5
0

35
30
25
20
15
10
5
0

35
30
25
20
15
10
5
0

35
30
25
20
15
10
5
0

35
30
25
20
15
10
5
0

We now have a cumulative frequency plot
35
30
25
20
15
10
5
0

Let’s try
to apply
this to
toxicology
testing

Dose (mg/kg) No of deaths No of test % deaths
animals
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
died
Dose (mg/kg) No of deaths No of test % deaths
animals
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
died
Dose (mg/kg) No of deaths No of test % deaths
animals
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
Dose (mg/kg) No of deaths No of test % deaths
animals
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
Dose (mg/kg) No of deaths No of test % deaths
animals
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
Dose (mg/kg) No of deaths No of test % deaths
animals
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
Dose (mg/kg) No of deaths No of test % deaths
animals
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
Dose (mg/kg) No of deaths No of test % deaths
animals
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
mg/kg
Dose (mg/kg) No of deaths No of test % deaths
animals
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 doses
Dose (mg/kg) No of deaths No of test % deaths
animals
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

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

No. of deaths
0
Administered dose mg/kg body weight

25
20
15
No. of deaths
10
5
0
5 10 25
Administered dose mg/kg body weight

25
20
15
No. of deaths
10
5
0
5 10 25 50
Administered dose mg/kg body weight

25
20
15
No. of deaths
10
5
0
5 10 25 50 100
Administered dose mg/kg body weight

25
20
15
No. of deaths
10
5
0
5 10 25 50 100 250
Administered dose mg/kg body weight

25
20
15
No. of deaths
10
5
0
5 10 25 50 100 250 500
Administered dose mg/kg body weight

25
20
15
No. of deaths
10
5
0
5 10 25 50 100 250 500 1000
Administered dose mg/kg body weight

Plot of the % deaths against administered dose (as an X Y Plot)
120
100
80
%. of animals dead
60
40
20
0
0 200 400 600 800 1000 1200
Administered dose mg/kg body weight

Plotted on a logarithmic scale
120
100
80
%. of animals dead
60
40
20
0
1 10 100 1000
Administered dose mg/kg body weight

We can now determine the LD50
120
100
80
%. of animals dead
60
40
20
0
1 10 100 1000
Administered dose mg/kg body weight

120
100
80
%. of animals dead
60
40
20
0
1 10 100 1000
Administered dose mg/kg body weight

120
100
80
%. of animals dead
60
40
20
0
1 10 100 1000
Administered dose mg/kg body weight

LD50
120
100
80
%. of animals dead
60
40
20
LD50
0
1 10 100 1000
Administered dose mg/kg body weight

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