# Find the median body mass index (BMI) for females in the United States of America.

Extracts from this document...

Introduction

The table below shows the median body mass index (BMI) for females in the United States of America (USA) for the year 2000 (source www.cdc.gov).

Age (yrs) | BMI |

2 | 16.4 |

3 | 15.7 |

4 | 15.3 |

5 | 15.2 |

6 | 15.21 |

7 | 15.4 |

8 | 15.8 |

9 | 16.3 |

10 | 16.8 |

11 | 17.5 |

12 | 18.18 |

13 | 18.7 |

14 | 19.36 |

15 | 19.88 |

16 | 20.4 |

17 | 20.85 |

18 | 21.22 |

19 | 21.6 |

20 | 21.65 |

Using the two variables above in the table, the independent variable (age), and the dependent variable BMI which has the units kg.m-2. The graph below was generated using Microsoft Excel 2007.

Using a GDC, the data was fitted to a set of parameters, a screen shot of these parameters can be seen below:

It can be seenfrom the plot that one possible mathematical fit for the graph is a sine function. This is apparent because it has the same periodic nature (it appears to be ¾ of a period of a sine curve) as a sine function. The next step in modelling a generic sine function to the data in the plot above, and applying a number of changes to f(x) = sin(x) curve. There are quite a number of calculations that are needed to acquire an accurate model of the data. Each aspect of the graph is investigated separately, for example vertical shift.

The graph below shows you the general function of a sine equation.

y=Sin(x)

Middle

Now the model becomes, f(x) =

The model function is now compared to the plot of the original data table (BMI).

The graph below shows the model compared to the actual data using the GDC.

This can also be seen using excel, the model was applied to the same independent variable values (ages), the results can be seen in the following diagram.

Age (yrs) | BMI | Model |

2 | 16.4 | 15.82473 |

3 | 15.7 | 15.48645 |

4 | 15.3 | 15.27683 |

5 | 15.2 | 15.20501 |

6 | 15.21 | 15.27413 |

7 | 15.4 | 15.48119 |

8 | 15.8 | 15.81712 |

9 | 16.3 | 16.26725 |

10 | 16.8 | 16.8119 |

11 | 17.5 | 17.42726 |

12 | 18.18 | 18.08646 |

13 | 18.7 | 18.76066 |

14 | 19.36 | 19.42042 |

15 | 19.88 | 20.03689 |

16 | 20.4 | 20.58313 |

17 | 20.85 | 21.03527 |

18 | 21.22 | 21.37354 |

19 | 21.6 | 21.58317 |

20 | 21.65 | 21.65499 |

As you can see there is a slight difference between the model graph and the original data graph.

They both have similar curve nature. The model function has a constant up and down curve while the original data graph has one curve and then it remains constant.

By using excel we were able to find another graph that fits the data. This is a polynomial function.

The equation for the function is X5+X4+X3 +X2+X1+X by adding your data in the graph we were able to find that the polynomial function has the best fit to the graph.

The graph below shows the function compared to the model:

The difference between the sin function and the polynomial function was that the sin function has constant curve movement; however the polynomial function had the best fit.

Conclusion

Calculating the different aspects of the sine function will help to obtain best fit for the data above:

Vertical shift

d = 18.4

f(x)= sin(x)+18.4

Amplitude

a = 3.2

f(x)=3.2sin(x)+18.4

Period

B =

F(x)=3.2sin(x)+18.4

Horizontal shift

b is known to be

=12.5

-c = 12.5 x

-c =2.61

C= -2.61

F(x)=3.2sin(x-2.61)+18.4

Using excel the model was applied to the same independent variable values (ages), the results can be seen in the following diagram.

Age (yrs) | BMI | model |

2 | 16.5 | 15.82473 |

3 | 15.6 | 15.48645 |

4 | 15.4 | 15.27683 |

5 | 15.2 | 15.20501 |

6 | 15.4 | 15.27413 |

7 | 15.5 | 15.48119 |

8 | 15.75 | 15.81712 |

9 | 16.4 | 16.26725 |

10 | 16.8 | 16.8119 |

11 | 17.5 | 17.42726 |

12 | 18.1 | 18.08646 |

13 | 18.6 | 18.76066 |

14 | 19.3 | 19.42042 |

15 | 20 | 20.03689 |

16 | 20.4 | 20.58313 |

17 | 20.9 | 21.03527 |

18 | 21.3 | 21.37354 |

19 | 21.5 | 21.58317 |

20 | 21.6 | 21.65499 |

The graph below shows the model compared to the actual data using the GDC.

-There is a slight difference between the model graph and body mass index (BMI) of young Australian woman. They have a similar curve nature for a specific period before the sine function changes.

-The sine function has a constant up and down curve while the original data graph has one curve and then it remains constant.

-The BMI of the Australian females is very similar to the BMI of US females from the age of 2 -20.

-Until this point the sine function has been able to calculate the BMI of different woman in different countries, however if you apply more tests to the function with different countries and ages then you will see slight limitations to the graph which would require you to either recalculate the equation or use other functions.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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