Mean and median
The mean of a sample: \((x_1,x_2,...,x_n)\) of size \(n\)
of a quantitative variable is computed as:
$$\bar{x}=\frac{\sum_{\forall i}x_i}{n}$$
The median is the value that separates the sample in two, so that the 50% of the
observation are below the median and the other 50% are above the median. In this exercise
you can compare the mean and the median of a sample and the different statistics related
to a boxplot. The residual for each observation in the value
\((x_i-\bar{x})\). Here, the residual for the minimum value is shown.
In the variance option, we will examine this concept for all the points
in the sample.
Meaning of the variance
The sample variance is computed as:
$$var(x)=\frac{\sum_{\forall i} (x_i-\bar{x})^2}{n-1}$$
The term \((x_i-\bar{x})\) is known as the residual and computes the
deviation of an observation with respect the mean of the sample. The
variance is the average of the squares of these residuals.
In this application you can fix the population mean and standard deviation
and obtain a sample assuming the biomarker values are normally distributed.
In the panel, you can select a data point and see the residual value.
Quantiles and percentiles
The percentile \(x_q\) is the variable value for which a \(q\%\) of the
values are below \(x_q\). For instance, if in a population the 95% of the
males have a value of a biomarker below 3.24mg/ml, then this value is
the percentile 95% for this biomarker. The quantiles are the
25%, 50%, and 75% percentiles. In this exercise, we show the sample quantiles
as computed in the boxplot. We also include a statistic summary of the groups
resulting from deviding the sample by quantiles. Finally, you can obtain a
given percentile for the sample using 9 different methods. For small samples,
the different methods produce different results.
Reference intervals
A reference interval of probability \((1-\alpha)\) is an interval \((a,b)\)
defined as:
$$P(X\leq a)=\alpha/2$$
$$P(X\leq b)=1-\alpha/2$$
Thus, \(a\) is the \((\alpha/2)\) percentile, and \(b\) is the \((1-\alpha/2)\)
percentile. In this exercise, you can obtain the reference interval estimated from the sample
using 9 different methods. The actual values, assuming the sample comes from a
normal distribution are shown in blue. The confidence interval for the reference interval
can also be plotted. For small samples, the results are not precise. Try to increase the sample
size to obtained a good result. If the intervals are not shown, you should change
the range of the plot using the slider.