Plotting Frequency Distributions

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


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


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


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


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


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


Cumulative



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


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


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

April


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

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

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


% 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


% 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


% 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

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 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


No.

of

death

s

5 10 25 500

5

10

15

20

25


No.

of

death

s

5 10 25 50 1000

5

10

15

20

25


No.

of

death

s

5 10 25 50 100 2500

5

10

15

20

25


No.

of

death

s

5 10 25 50 100 250 5000

5

10

15

20

25


No.

of

death

s

5 10 25 50 100 250 500 10000

5

10

15

20

25


No.

of

death

s

0 200 400 600 800 1000 12000

20

40

60

80

100

120


%.

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


%.

of

an

imals

dead

Plotted on a logarithmic scale

1 10 100 10000

20

40

60

80

100

120


%.

of

an

imals

dead

We can now determine the LD50

1 10 100 10000

20

40

60

80

100

120


%.

of

an

imals

dead

1 10 100 10000

20

40

60

80

100

120


%.

of

an

imals

dead

LD50

LD50

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Plotting Frequency Distributions

Health & Medicine

Transcript of Plotting Frequency Distributions