The document presents data on the number of pupils born in each month in a class. It then shows how to create a histogram and cumulative frequency plot of the data. Specifically: - Data is given on the number of pupils born each month, ranging from 1 in March to 5 in May. - A histogram is created to visualize the frequency of births by month. - A cumulative frequency plot is then created by summing the frequencies of births from previous months to show the running total of births by month. - This cumulative approach is demonstrated on sample toxicology data, showing the cumulative number of deaths from different dose levels of a toxin.

1. Mike Slater

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

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

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

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

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

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

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

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

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

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

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

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

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

18. Now there is another way of looking at our data

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

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

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

54. 35
30
25
20
15
10
5
0

55. 35
30
25
20
15
10
5
0

56. 35
30
25
20
15
10
5
0

57. 35
30
25
20
15
10
5
0

58. 35
30
25
20
15
10
5
0

59. 35
30
25
20
15
10
5
0

60. 35
30
25
20
15
10
5
0

61. 35
30
25
20
15
10
5
0

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

63. Let’s try
to apply
this to
toxicology
testing

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

75. So we have CUMMULATIVE data

76. If we plot this data we have a CUMMULATIVE
frequency plot

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

78. 25
20
15
No. of deaths
10
5
0
5 10 25
Administered dose mg/kg body weight

79. 25
20
15
No. of deaths
10
5
0
5 10 25 50
Administered dose mg/kg body weight

80. 25
20
15
No. of deaths
10
5
0
5 10 25 50 100
Administered dose mg/kg body weight

81. 25
20
15
No. of deaths
10
5
0
5 10 25 50 100 250
Administered dose mg/kg body weight

82. 25
20
15
No. of deaths
10
5
0
5 10 25 50 100 250 500
Administered dose mg/kg body weight

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

88. 120
100
80
%. of animals dead
60
40
20
0
1 10 100 1000
Administered dose mg/kg body weight

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

90. http://www.diamondenv.co.uk
http://occhygiene.blogspot.com/