Mathematical tools for natural sciences

Once we have a data set, we try to identify any feature in the data. By *feature* we mean the following: among the range of values the data points have, do all the values occur in equal proportion or certain values occur more often than the others? To anwser this quation, the * frequency distribution * of the data has to be created. The number of times a given value repeats in a data is called its * frequency of occurance *

We will create the frequency distribution of the following data set which contains the age in years of the children playing in a park on a particular sunday evening:

5 7 6 7 6 5 6 6 7 8 6 6 8 5 8 5 5 5 5 8 2 8 8 5 8 5 7 5 9 2 7 5 5 8 6 6 5 8 1 2 5 8 6 8 8 5 9 7 7 6 7 6 5 7 9

From this data, a table with the frequency of occurance of each age is created. This is the number of children of given age playing inside the park. Divide this frequency by total number of children (55) to get the * relative frequency * of each value. This fraction is actually the * experimental probability* or the * empirical probability * for randomly selecting a child of given age from the data. The probability is defined as,

\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~p = \dfrac{Frequency~of~a~data~value}{total~number~of~data~points} \)

age(years) | frequency | empirical probability |
---|---|---|

2 | 3 | 0.0545 |

3 | 5 | 0.0909 |

4 | 8 | 0.1454 |

5 | 11 | 0.2 |

6 | 11 | 0.2 |

7 | 9 | 0.1636 |

8 | 4 | 0.0727 |

9 | 3 | 0.05454 |

10 | 1 | 0.01818 |

n = sum = 55 |

The above table gives the frequency distribution and the empirical probability distribution of the age of children in the park. The plots of these two distributions is shown below:

The frequency and the probability distributions reveal interesting features about the data which may not be obvious when we stare just at the numbers. For example, we realize that on the given day, more children in the age group of 4-7 years visited the park when compared to older or younger children.

When a data set has large number of discrete values span over a wide range, it is combursome to look
at the frequency plot. It is more convenient to divide the data set into * intervals of equal width* and
count the frequency of data points that contrubute to each of the intervals. This is called binning,
and the plot showing the frequency of each bin agains the bin width is called a * histogram *.

As an example, let us look at the following imaginary data set on the age(in years) of the male patients visiting a clinic over a period of three days:

22, 7, 10, 32, 52, 53, 54, 56, 68, 59, 34,8, 24, 14, 62, 64, 66, 81, 84, 26, 7, 16, 17, 33, 36, 28, 30, 29, 70, 72, 73, 39, 40, 42, 43, 44, 45, 46, 48, 50

There are 40 data points, ranging from a minimum value of 7 years to a maximum value of 84 years.

We can choose, for example, bins of width 10 years starting from 0 ending in 100. ie., the bins are
as follows: 0 – 10, 10 – 20, 20 – 30, and so on upto 90 – 100.

For each bin, we include the data points whose values are more than or equal to the lower range and
less than the upper range. We call this “ left included and right excluded ”. There are no rigid rules,
but this is the general convention.

For example, the following 3 data points come in the range 0 – 10 of first bin : 7,8,7. So, the frequency
in this bin is 3.

Similarly the data points 10,14,16,17 come in the range 10 – 20 of second bin. So, the
frequency of this bin is 4.

The frequency of every bin can be computed like this and a frequency table can be made like before:

age bin (years) | frequency | empirical probability |
---|---|---|

0-10 | 3 | 0.075 |

10-20 | 4 | 0.1 |

20-30 | 5 | 0.125 |

30-40 | 6 | 0.15 |

40-50 | 7 | 0.175 |

50-60 | 6 | 0.15 |

60-70 | 4 | 0.1 |

70-80 | 3 | 0.075 |

80-90 | 2 | 0.05 |

sum = 40 |

The frequency and empirical probability historgrams of the above data are shown below:

While plotting this histogram, we have divided the age range into 5 bins of 20 years width. We may choose some other range based on other considerations.

756.6, 792.1, 891.9, 765.2, 718.5, 902.3, 780.3, 927.7, 927.9, 940.2, 743.5, 829.8, 784.3, 964.3, 794.7, 1056.1, 1124.2, 736.5, 979.5, 781.6, 986.4, 957.5, 901.0, 972.8, 905.9, 983.7, 970.7, 876.0, 864.6, 876.4, 962.3, 899.7, 1061.0, 980.3, 919.5, 993.9, 903.1, 1005.0, 860.5, 905.0, 813.5, 1040.5, 951.3, 958.7, 949.7, 983.4, 975.5, 977.2, 940.8, 923.2, 749.2, 827.8, 983.4, 914.4, 962.0, 987.8, 898.1, 1012.9, 1036.7, 930.0, 1078.2, 893.6, 912.2, 1031.4, 738.3, 779.8, 884.1, 791.6, 883.3, 998.7, 885.7, 697.4, 956.1, 798.8, 1011.4, 901.5, 911.3, 965.2, 724.8, 912.1, 887.1, 767.4, 1001.5, 859.8, 832.5, 769.9, 774.6, 1094.1, 920.0, 972.3, 828.3, 831.7, 905.7, 1001.2, 900.3, 935.1, 927.3, 943.1, 863.1, 833.7, 821.9, 737.3, 919.5, 774.2, 874.3, 889.3, 943.0, 877.7, 698.2, 911.1, 901.2, 823.6, 937.4

As with the previuos example, we divide the data into optimal bins and get the frequency and probability distributions of the data, as presented in the following table and figure:

rainfall bin (mm) | frequency | empirical probability |
---|---|---|

650-700 | 2 | 0.0176 |

700-750 | 7 | 0.0619 |

750-800 | 14 | 0.1239 |

800-850 | 9 | 0.0796 |

850-900 | 17 | 0.1504 |

900-950 | 29 | 0.2566 |

950-1000 | 22 | 0.1947 |

1000-1050 | 8 | 0.0708 |

1050-1100 | 4 | 0.03539 |

1100-1150 | 0.0088 | |

sum = 113 |

When we represent the distribution of an observed data set with a frequency histogram, the number of bins we choose for the histogram is decided by the data size. For a given data set, as the number of bins increases, the number of data points in the individual bins decrease. So we need large number of data points to maintain statistically significant number of events per bin. When we have more and more data points coming, we can increase the number of bins and reduce the bin width.

In the limit of a very large data set (the number n of data points tends to very large value, towards infinity), the frequency histogram can have an infinitely large number of bins, each having a width approaching zero. In this case, **the frequency and the probability distributions tend to be continuous curves. **

We will illustrate this idea using simulated data points. Using some mathematical function (will be described in later sections), we can randomly draw data points from a Gaussian distribution of certain mean and standard deviation. We plot the probability histograms of 4 data subsets drawn from this Gaussian. The four histograms have *increasing sizes and decreasing bin widths*. See the set of histograms below:

In the four plots given above, we see that as the number of bins increases, the bin width decrease and the probability histogram starts looking like a continuous curve (shown as red line in the fourth plot). We can think other way around. ** In nature, some populations have continuous distribution. The values of the data points are continuous such that in any given range we can have infinite number of data points. Thus, we can have values like 12.45, 12.4532, 12.453219, ....
Such a data can only be represented as a continuous distribution, described by a continuous curve**.

To understand this point, we will draw the probability distribution of a continuous variable x represented by some mathematical function P(x), as shown here:

In a discrete probability distribution, the variable x takes discrete values, and probability of all these discrete values sum to total probability 1.

In a continuous probability distribution, the variable x can take infinite number of values in a given range. Since the probabilities of all these infinite values should sum to one, it is clear that the probability of individual x values should be infinitesmally small. Therefore, in a continuous distribution, we do not consider the probability of observing a given x value. ** We consider the probability of observing a value within a unit interval around x, called as probability density P(x) **.

If P(x) is the probability per unit interval around x (probability density), then ** P(x)dx ** gives the
probability of observing a value within an interval dx around the value x.

As depicted in the diagram above, P(x)dx is the area of the probability density curve under a strip of width dx around x. This is equal to the probability of sampling a value within width dx around x.

What is the probability of observing an x value between x=x1 and x=x2?. This is given by the area under the curve between x=a and x=b. This area is obtained by the integral \(\int_{x1}^{x2} P(x)dx \)

** To summarize : **

\( P(x) \) = Probability_density at x = Probability of observing a value within an unit interval around x

\( P(x)dx \) = Probability of observing a value

\(\int_a^b P(x)dx \) = Area under the curve from x=a to x = b.

\( ~~~~~~~~~~~~~~\) = Probability of observing a value in the interval [a, b]

The data points from an experiment are assumed to have been randomly sampled from a * parent population (simply called "population")*. Each one of these populations, whether discrete or continuous, is represented by a probability distribution which has a mathematical expression.

** The question is, for a given population we sample, who gives us the mathematical expression of the probability or probability density distribution?
Mostly they are derived using the fundamental mathematics that governs the underlying phenomena. We will learn about them in the coming sections.**

- Binomial distribution
- Poisson distribution
- Hypergeometric distribution
- Geometric distribution
- Negative binomial distributionb

- Gaussian distribution
- Uniform distribution
- Exponential distribution
- Gamma distribution
- Chi-square distribution
- Student's t distribution
- F distribution