Statistics are often taught in school by and for people who like Mathematics. As a consequence, in those class emphasis is put on leaning equations, solving calculus problems and creating mathematics models instead of building an intuition for probabilistic problems. But, if you read this, you know a bit of R programming and have access to a computer that is really good at computing stuff! So let’s learn how we can tackle useful statistic problems by writing simple R query and how to think in probabilistic terms.
In the last exercise set we’ve seen that random variable can be described by mathematical functions called probability density and that when we know which one describe a particular random process we can use it to compute the probability of realization of a given event. We have also seen how to use an histogram and an ECDF plot to identify which function express the random variable. Today, we will see which mathematical properties of those function we can compute to help us find the probability density who fit a sample. Those properties are called statistics and our job today is to estimate the real value of those properties by using a small sample of data.
Answers to the exercises are available here.
The most commonly used statistics is the mean, which is the center of mass of the distribution, i.e. the point on the x axis where the weighted relative position of each observation sum to zero. For example, draw the density of a standard normal distribution and add to the plot a vertical line to indicate the mean of this distribution. Then, draw another plot, but this time of an exponential distribution with a rate of 1 and his mean.
From the density plot of the standard normal distribution we can see how the mean represent the center of mass of the distribution: the normal distribution is symmetric, so the mean is in the center of the plot of the function. The exponential function is not symmetric, in this case the mean is the point where all the points with a small y value, at the right of the mean on the plot, counterbalance the few points with a high y value at the left of the mean. Since the value of the mean is at the center of the distribution, we often use the mean to represent a typical value of a probability distribution. The mean also give us the ability to put a number on the location of a probability distribution on the axis.
In practice, we don’t have access to the probability density function of a random variable and can’t compute directly the mean of the distribution. We must estimate it using a sample of observations of that random variable. Since it’s random, all samples will be different and our estimations of the mean, will all be different.
Generate 500 points from an exponential distribution with a rate of 0.5. Draw the histogram of the sample and compute the sample mean of this distribution. Then write a function that repeat this process for n iterations, store the sample mean in a vector and return this vector. Use this function to compute 10,000 sample means, plot the histogram of the sample means and compute the mean of those estimations.
From the histogram of the sample mean, we can see that the estimations follow a normal distribution centered around the real value of the mean. We can use this fact to compute the interval which have a certain probability of containing the real value of the mean. This interval is called the confidence interval of the estimate and the probability that this interval contain the real mean of the distribution is called the confidence level. In the next exercise set, we will see methods to compute this interval directly from a sample without knowing the probability density function of the random variable.
quantile() function to compute the 2.5 and 97.5 percentile from the sample of estimations of the mean, then use the
t.test() function to compute the confidence interval with a level of 95% of the original distribution and compare those values.
Load this dataset and use the
t.test() function to compute the confidence interval of the mean for both variables with a level of 95%. Does those random variables seems to follow distributions who have the same means?
We see that the confidence intervals doesn’t overlap. This is an indication that the real value of the mean of the first variable is not in the same interval as the distribution mean of the second variable. As a consequence, we can safely suppose than both mean are different and that they don’t have the same probability distribution.
Another useful statistics is the variance. This statistic is an indication of how the data are spread around the mean. So if two distributions have the same mean, the one with the smallest variance has the most homogeneous value, while the one with the highest variance has more small and high value far from the mean. A related statistics is the standard deviation, which is defined as the square root of the variance.
Draw the density of a standard normal distribution and of a normal distribution of mean equal to zero and with a standard deviation of 5 to see the effect of a change of variance on a density.
In the case of the variance, we cannot directly compute the confidence interval without making assumption on the type of distribution the sample come from or use some fancy method we will introduce in the next exercise set. Luckly for us we can use the
var.test() function to verify if the variances are equal. Use this function on the dataset of exercise 4 three time, once with the alternative parameter set to “two.sided”, then to “less” and finally to “greater”. What is the signification of the three test?
If the mean is a good representation of the typical value of a random variable defined by a density, this statistics can be skew by outliers. When a sample has outlier a better statistics to use is the median, which is the value that separate the range of observations that can be generated by a random variable in two equal parts.
Generate 200 points from a log-normal distribution with a parameter
meanlog = 0 and
sdlog = 0.5. Then plot the histogram of those points and represent the mean and the median of this sample by two vertical lines.
The median is a special case of a more general statistics called quantile, which are cutpoints dividing the domain of a probability density function into sub-interval containing the same amount of observations. So the 2-quantile is the median, since this statistics separate the domain of a probability distribution in two sub-interval containing 50% of the observations. Other quantile statistics often used are the 4-quantile, called quartile, which are the values on the domain of a probability distribution that separates it in four sub-interval containing 25% of the observations and the 100-quantile, called percentille, which are the values that separate this domain in 100 part containing 1% of the observations.
Compute the median, the quantile and the 5 and 95% percentile on the variables of the dataset of exercise 4. Then compute the interquartile range which is the difference between the 25% and the 75% quartile. Does those statistics suggest that the two samples have the same distribution?
Another statistics that can be used to differentiate two probability distribution is the skewness. As his name imply, the skewness is a measure of how much there is an imbalance between the observations at the right of the mean and at the left of the mean. A negative skewness indicate that the distribution is skew to the left, a positive value indicate that the distribution is skew to the right and a skewness of zero tell us that the distribution is perfectly symmetric.
moment package and use the
skewness() function to compute the skewness of three samples you must create:
- 150 points sample from a standard normal distribution
- 1000 points sample from a standard normal distribution
- 200 points sample from a exponential distribution with a rate of 5
The last statistic we will use today is the kurtosis, which describe the general shape of the probability distribution. When the kurtosis is greater than zero, the probability distribution has heavy tail and a pointy shape. Both of those characteristics are proportional to the magnitude of the kurtosis. If the kurtosis is less than zero, the distribution has a more regular shape with light tails. When this statistic has a value of zero, the distribution’s shape look a lot like the normal distribution.
kurtosis() function to compute the kurtosis of those samples:
- 500 points sample from a standard normal distribution
- 500 points sample from a exponential distribution with a rate of 5
- 500 points sample from uniform distribution