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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
month
January 2February 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
2 pupils had birthdays in January
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
2 pupils had birthdays in January
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
2 pupils had birthdays in February
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
1 pupils had a birthdays in March
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
4 pupils had birthdays in January
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
No of pupils with birthday in a given month
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
1
2
3
4
5
6
This is a frequency density plot
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 that month
Cumulative
January 2February 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2February 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
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
this
Month No of pupils born that month
Cumulative
January 2 2February 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
For the other months add the data from the
month in question
Month No of pupils born that month
Cumulative
January 2 2February 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
For the other months add the data from the
month in question
Month No of pupils born that month
Cumulative
January 2 2February 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
To the cumulative figure from the previous month
Month No of pupils born that month
Cumulative
January 2 2February 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
To the cumulative figure from the previous month
Month No of pupils born that month
Cumulative
January 2 2February 2 2 + 2March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 2 + 2 = 4March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1April 4May 5June 5July 4August 3September 2October 2November 1December 1
Now do the same for March
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 1 + 4April 4May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 1 + 4 = 5April 4May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 5April 4May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 5April 4May 5June 5July 4August 3September 2October 2November 1December 1
Now do the same for
April
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 5April 4 4 + 5May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 5April 4 4 + 5 = 9May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 5April 4 9May 5June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
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
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 5April 4 9May 5 14June 5July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4 23August 3September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4 23August 3 26September 2October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4 23August 3 26September 2 28October 2November 1December 1
Month No of pupils born that month
Cumulative
January 2 2February 2 4March 1 5April 4 9May 5 14June 5 19July 4 23August 3 26September 2 28October 2 30November 1December 1
Month No of pupils born that month
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
Month No of pupils born that month
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
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
In January 2 pupils had birthdays
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
In the period January to February 2 + 2 = 4 pupils had birthdays(2 in January and 2 in February)
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
In the period January to March 5pupils had birthdays(2 in January + 2 in February + 1 in March)
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
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
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
We now have a cumulative frequency plot
Janu
ary
Febru
ary
Mar
ch
April
May
June
Ju
ly
Augus
t
Sept
embe
r
Oct
ober
Nov
embe
r
Decem
ber
0
5
10
15
20
25
30
35
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
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
At 5 mg/kg no animals
died
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
At 10 mg/kg no animals
died
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
At 25 mg/kg
3 animals died
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
At 50 mg/kg
7 animals died
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
But some of these
would have died at a
lower dose
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
3 animals died at
25 mg/kg
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
So an EXTRA 4
died at the higher dose
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
15 animals died at 100
mg/kg
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
So an EXTRA 8 died than at lower
doses
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
5
10
15
20
25
Administered dose mg/kg body weight
No.
of
death
s
5 10 250
5
10
15
20
25
Administered dose mg/kg body weight
No.
of
death
s
5 10 25 500
5
10
15
20
25
Administered dose mg/kg body weight
No.
of
death
s
5 10 25 50 1000
5
10
15
20
25
Administered dose mg/kg body weight
No.
of
death
s
5 10 25 50 100 2500
5
10
15
20
25
Administered dose mg/kg body weight
No.
of
death
s
5 10 25 50 100 250 5000
5
10
15
20
25
Administered dose mg/kg body weight
No.
of
death
s
5 10 25 50 100 250 500 10000
5
10
15
20
25
Administered dose mg/kg body weight
No.
of
death
s
0 200 400 600 800 1000 12000
20
40
60
80
100
120
Administered dose mg/kg body weight
%.
of
an
imals
dead
Plot of the % deaths against administered dose (as an X Y Plot)
1 10 100 10000
20
40
60
80
100
120
Administered dose mg/kg body weight
%.
of
an
imals
dead
Plotted on a logarithmic scale
1 10 100 10000
20
40
60
80
100
120
Administered dose mg/kg body weight
%.
of
an
imals
dead
We can now determine the LD50
1 10 100 10000
20
40
60
80
100
120
Administered dose mg/kg body weight
%.
of
an
imals
dead
1 10 100 10000
20
40
60
80
100
120
Administered dose mg/kg body weight
%.
of
an
imals
dead
1 10 100 10000
20
40
60
80
100
120
Administered dose mg/kg body weight
%.
of
an
imals
dead
LD50
LD50
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