Polygon and histogram

For clarity, various graphs of statistical distribution are constructed, in particular, a polygon and a histogram.

Frequency range is a polyline, the segments of which connect the points . To build a range of frequencies on the x-axis, lay the options , and on the y-axis - the corresponding frequency and connect the dots straight line segments.

The range of relative frequencies is constructed similarly, except that relative frequencies are plotted on the ordinate axis .

In the case of a continuous feature, a histogram is built, for which the interval in which all the observed values ​​of the feature are enclosed is divided into several partial intervals of length h and found for each partial interval - the sum of frequencies of an option that fell into the i – th interval.

The histogram of frequencies is called a step shape consisting of rectangles, the bases of which are partial intervals of length h, and the heights are equal to the ratio . To construct a histogram of frequencies on the x-axis lay partial intervals, and above them spend the segments parallel to the x-axis at a distance (height) . The area of ​​the i – th rectangle is - the sum of frequencies is a variant of the i – о interval, therefore the area of ​​the frequency histogram is equal to the sum of all frequencies, i.e. sample size.

In the case of relative frequency histograms, relative frequencies are plotted along the ordinate axis , on the abscissa axis - partial intervals, over them spend the segments parallel to the abscissa axis at the height . The area of ​​the i – th rectangle is equal to the relative frequency trapped in the i – th interval. Therefore, the area of ​​the histogram of relative frequencies is equal to the sum of all relative frequencies, that is, one.

For clarity, build various graphs of statistical distribution.

According to a discrete variation series, a range of frequencies or relative frequencies are built.

Frequency range is a broken line whose segments connect the points ( x 1 ; n 1 ), ( x 2 ; n 2 ), ..., ( x k ; n k ). To build a range of frequencies on the x- axis lay the options x i , and on the y-axis - the corresponding frequency n i . Points ( x i ; n i ) are connected by straight lines and receive a range of frequencies (Fig. 1).

The polygon of relative frequencies is called a broken line, the segments of which connect the points ( x 1 ; W 1 ), ( x 2 ; W 2 ), ..., ( x k ; W k ). To build a polygon of relative frequencies on the x- axis lay options x i , and on the y-axis - the corresponding relative frequencies W i . Points ( x i ; W i ) are connected by straight line segments and receive a range of relative frequencies.

In the case of a continuous feature, it is advisable to build a histogram .

The histogram of frequencies is called a stepped figure, consisting of rectangles, the bases of which are partial intervals of length h , and the heights are equal to the ratio n i / h (frequency density).

To construct a histogram of frequencies on the x-axis lay partial intervals, and on them spend the segments parallel to the x-axis at a distance of n i / h .

The area of ​​the i - th partial rectangle is equal to hn i / h = n i - the sum of the frequencies of the variant of the i - th interval; therefore, the area of ​​the frequency histogram is equal to the sum of all frequencies, i.e. sample size .

The histogram of relative frequencies is called a step shape, consisting of rectangles, the bases of which are partial intervals of length h , and the heights are equal to the ratio W i / h (relative frequency density).

To construct a histogram of relative frequencies, partial intervals are laid on the abscissa axis, and over them are carried out segments parallel to the abscissa axis at distance W i / h (Fig. 2).

The area of ​​the i - th partial rectangle is equal to hW i / h = W i - to the relative frequency of the variant in the i - th interval. Consequently, the area of ​​the histogram of relative frequencies is equal to the sum of all relative frequencies, i.e. unit .