Mike Slater
Here is some data – the number of pupils in
a class with birthdays in each month
     Month        No of pupils born that ...
Let’s plot a histogram for this data
     Month          No of pupils born that month
     January                       2...
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     ...
Month       No of pupils born   Cumulative
            that month
January              2
February             2
March     ...
Month       No of pupils born   Cumulative
            that month
January              2              2+ 0 = 2
February   ...
Month       No of pupils born   Cumulative
            that month
January              2                  2
February      ...
Month       No of pupils born   Cumulative
            that month
January              2                  2
February      ...
Month       No of pupils born   Cumulative
            that month
January              2                  2
February      ...
Month       No of pupils born   Cumulative
            that month
January              2                  2
February      ...
Month       No of pupils born   Cumulative
            that month
January              2                  2
February      ...
Month       No of pupils born   Cumulative
            that month
January              2                2
February        ...
Month       No of pupils born   Cumulative
            that month
January              2                2
February        ...
Month       No of pupils born   Cumulative
            that month
January              2                  2
February      ...
Month       No of pupils born   Cumulative
            that month
January              2                  2
February      ...
Month       No of pupils born   Cumulative
            that month
January              2                2
February        ...
Month       No of pupils born   Cumulative
            that month
January              2               2
February         ...
Month       No of pupils born   Cumulative
            that month
January              2                  2
February      ...
Month       No of pupils born   Cumulative
            that month
January              2                  2
February      ...
Month       No of pupils born   Cumulative
            that month
January              2                2
February        ...
Month       No of pupils born   Cumulative
            that month
January              2                2
February        ...
Month       No of pupils born   Cumulative
            that month
January              2                  2
February      ...
Month       No of pupils born   Cumulative
            that month
January              2                  2
February      ...
Month       No of pupils born   Cumulative
            that month
January              2                 2
February       ...
Month       No of pupils born   Cumulative
            that month
January              2                 2
February       ...
Month       No of pupils born   Cumulative
            that month
January              2                 2
February       ...
Month       No of pupils born   Cumulative
            that month
January              2                 2
February       ...
Month       No of pupils born   Cumulative
            that month
January              2                 2
February       ...
Month       No of pupils born   Cumulative
            that month
January              2                 2
February       ...
Month       No of pupils born   Cumulative
            that month
January              2                 2
February       ...
Month       No of pupils born   Cumulative
            that month
January              2                 2
February       ...
Now let’s plot a chart for the cumulative data
      Month       No of pupils born   Cumulative
                  that mon...
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...
35
30
25
     In the period January to March 5pupils had birthdays
20   (2 in January + 2 in February + 1 in March)
15
10
...
35
30
     In the period January to April 9 pupils had birthdays
25   (2 in January + 2 in February + 1 in March + 4 in Ap...
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...
At 5 mg/kg
                              no animals
                                 died


Dose (mg/kg)   No of deaths   ...
At 10 mg/kg
                              no animals
                                  died


Dose (mg/kg)   No of deaths ...
At 25 mg/kg
                               3 animals
                                  died


Dose (mg/kg)   No of deaths ...
At 50 mg/kg
                               7 animals
                                  died


Dose (mg/kg)   No of deaths ...
But some of
                               these would
                              have died at a
                      ...
3 animals
                               died at
                              25 mg/kg


Dose (mg/kg)   No of deaths   No...
So an EXTRA
                              4 died at the
                               higher dose


Dose (mg/kg)   No of ...
15 animals
                              died at 100
                                mg/kg


Dose (mg/kg)   No of deaths  ...
So an EXTRA
                              8 died than at
                               lower doses


Dose (mg/kg)   No of...
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



             ...
25



                20



                15
No. of deaths




                10



                 5



             ...
25



                20



                15
No. of deaths




                10



                 5



             ...
25



                20



                15
No. of deaths




                10



                 5



             ...
25



                20



                15
No. of deaths




                10



                 5



             ...
25



                20



                15
No. of deaths




                10



                 5



             ...
Plot of the % deaths against administered dose (as an X Y Plot)
                     120


                     100


    ...
Plotted on a logarithmic scale
                     120


                     100


                     80
%. of animals...
We can now determine the LD50
                     120


                     100


                     80
%. of animals ...
120


                     100


                     80
%. of animals dead




                     60


                ...
120


                     100


                     80
%. of animals dead




                     60


                ...
LD50
                     120


                     100


                     80
%. of animals dead




                ...
http://www.diamondenv.co.uk
http://occhygiene.blogspot.com/
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Plotting Frequency Distributions

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Plotting Frequency Distributions

  1. 1. Mike Slater
  2. 2. 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
  3. 3. 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
  4. 4. No of pupils with birthday in a given month 1 2 pupils had birthdays in January 0
  5. 5. No of pupils with birthday in a given month 6 5 2 pupils had birthdays in January 4 3 2 1 0
  6. 6. No of pupils with birthday in a given month 6 5 2 pupils had birthdays in February 4 3 2 1 0
  7. 7. No of pupils with birthday in a given month 6 5 1 pupils had a birthdays in March 4 3 2 1 0
  8. 8. No of pupils with birthday in a given month 6 5 4 pupils had birthdays in January 4 3 2 1 0
  9. 9. No of pupils with birthday in a given month 6 5 4 3 2 1 0
  10. 10. No of pupils with birthday in a given month 6 5 4 3 2 1 0
  11. 11. No of pupils with birthday in a given month 6 5 4 3 2 1 0
  12. 12. No of pupils with birthday in a given month 6 5 4 3 2 1 0
  13. 13. No of pupils with birthday in a given month 6 5 4 3 2 1 0
  14. 14. No of pupils with birthday in a given month 6 5 4 3 2 1 0
  15. 15. No of pupils with birthday in a given month 6 5 4 3 2 1 0
  16. 16. No of pupils with birthday in a given month 6 5 4 3 2 1 0
  17. 17. This is a frequency density plot 6 5 4 3 2 1 0
  18. 18. Now there is another way of looking at our data
  19. 19. Now there is another way of looking at our data Lets look at it cummatively
  20. 20. 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
  21. 21. 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
  22. 22. 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
  23. 23. 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
  24. 24. 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
  25. 25. 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
  26. 26. 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
  27. 27. 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
  28. 28. 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
  29. 29. 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
  30. 30. 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
  31. 31. 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
  32. 32. 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
  33. 33. 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
  34. 34. 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
  35. 35. 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
  36. 36. 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
  37. 37. 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
  38. 38. 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
  39. 39. 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
  40. 40. 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
  41. 41. 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
  42. 42. 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
  43. 43. 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
  44. 44. 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
  45. 45. 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
  46. 46. 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
  47. 47. 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
  48. 48. 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
  49. 49. 35 30 25 20 15 In January 2 pupils had birthdays 10 5 0
  50. 50. 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
  51. 51. 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
  52. 52. 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
  53. 53. Now let’s do the same for the remaining months 35 30 25 20 15 10 5 0
  54. 54. 35 30 25 20 15 10 5 0
  55. 55. 35 30 25 20 15 10 5 0
  56. 56. 35 30 25 20 15 10 5 0
  57. 57. 35 30 25 20 15 10 5 0
  58. 58. 35 30 25 20 15 10 5 0
  59. 59. 35 30 25 20 15 10 5 0
  60. 60. 35 30 25 20 15 10 5 0
  61. 61. 35 30 25 20 15 10 5 0
  62. 62. We now have a cumulative frequency plot 35 30 25 20 15 10 5 0
  63. 63. Let’s try to apply this to toxicology testing
  64. 64. 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
  65. 65. 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
  66. 66. 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
  67. 67. 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
  68. 68. 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
  69. 69. 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
  70. 70. 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
  71. 71. 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
  72. 72. 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
  73. 73. 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
  74. 74. The deaths at each administered dose represent the TOTAL number of deaths caused by administering doses UP TO that dose
  75. 75. So we have CUMMULATIVE data
  76. 76. If we plot this data we have a CUMMULATIVE frequency plot
  77. 77. No. of deaths 0 Administered dose mg/kg body weight
  78. 78. 25 20 15 No. of deaths 10 5 0 5 10 25 Administered dose mg/kg body weight
  79. 79. 25 20 15 No. of deaths 10 5 0 5 10 25 50 Administered dose mg/kg body weight
  80. 80. 25 20 15 No. of deaths 10 5 0 5 10 25 50 100 Administered dose mg/kg body weight
  81. 81. 25 20 15 No. of deaths 10 5 0 5 10 25 50 100 250 Administered dose mg/kg body weight
  82. 82. 25 20 15 No. of deaths 10 5 0 5 10 25 50 100 250 500 Administered dose mg/kg body weight
  83. 83. 25 20 15 No. of deaths 10 5 0 5 10 25 50 100 250 500 1000 Administered dose mg/kg body weight
  84. 84. 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
  85. 85. 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
  86. 86. 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
  87. 87. 120 100 80 %. of animals dead 60 40 20 0 1 10 100 1000 Administered dose mg/kg body weight
  88. 88. 120 100 80 %. of animals dead 60 40 20 0 1 10 100 1000 Administered dose mg/kg body weight
  89. 89. LD50 120 100 80 %. of animals dead 60 40 20 LD50 0 1 10 100 1000 Administered dose mg/kg body weight
  90. 90. http://www.diamondenv.co.uk http://occhygiene.blogspot.com/

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