2st probability and statistical resource EQF2

statistics

training module

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statistical terms

frequency tables

Statistical graphs

Grouped data frequency tables

Histogram

Position parameters

Dispersion parameters

statistical terms

Population: The population is the set of elements on which a statistical study is carried out.

Sample: The sample is the part of the population from which the data are collected.

Individual: An individual is each element that is part of the sample or population.

statistical terms

Qualitative: cannot be expressed by a number

Quantitative: is expressed by a number.

Discrete: only takes isolated values.

Continuous: can take all values of an interval

Types of statistical variables

Statistical variable is the characteristic or property object of the study. It can be:

statistical terms

The stages of a statistical study are: *Selection of a representative sample. *Data collection. *Analysis of the data collected. *Drawing conclusions.

You should consider the following

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Frequency tables

From the data obtained in a statistical study, and to facilitate its study, a count is made and a table of frequencies is constructed. If the variable studied is qualitative or discrete quantitative, they are treated in the same way when representing them in a frequency table.

In a frequency table, the values taken by the statistical variable, xi, are represented with its associated frequencies: Absolute frequency, fi: is the number of times xi appears in the count. Relative frequency, hi: is the ratio between absolute frequency and total data number.hi = fi/N

The cumulative frequency of the last data matches the total number of data.

Accumulated absolute frequency, Fi: is the sum of absolute frequencies of values less than or equal to xi. Accumulated relative frequency, Hi: is the ratio between the accumulated frequency and the total number of data.Hi=Fi/N

Frequency tables

We want to study the number of daily tweets published by the 24 students of a class

Example

+info

To analyze the data, count and construct the frequency table:

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Through the frequency tables, information on a given statistical variable is organized, studied and obtained, but there is a more effective way of representing and collecting information: statistical graphs.

Statistical graphs

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If the midpoints of the upper ends of each bar are marked and joined, a new graph called the frequency polygon is obtained.

In a bar diagram each value is represented with a length bar proportional to its frequency. It is used for discrete qualitative and quantitative variables.

Bar diagram and frequency polygon

Bar Chart

Freqency range

Example

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In a sector diagram each value is represented by an amplitude sector proportional to its frequency. It is preferably used for qualitative variables.

Pie chart

The amplitude of each sector is the product of the relative frequency of the data by 360°: Sector amplitude (°) = hi.360°Sector graphs are useful only when the number of variable values is small

Example

Example

When a statistical variable is quantitative continuous or is discrete but takes many different values, the data are grouped in intervals to facilitate counting. To construct data frequency tables grouped in intervals, the mean value of each interval is usually taken as the representative value of each interval. This value is called the class mark, xi, of the range. All ranges of the same amplitude should be considered and their associated frequencies calculated in the same way as for discrete variables.

Grouped data frequency tables

Note the following: If an interval is closed with a bracket, the end is included in that interval. If closed with a parenthesis the end is not included. In the interval [0, 10), 0 is included and 10 is not..

Grouped data frequency tables

example

When data is grouped in intervals a new graph is used to represent them: the histogram. In a histogram each interval is represented by a rectangle whose base has the length of that interval and height proportional to its frequency. If we combine the class marks of each interval we obtain the frequency polygon.

Histogram

Fashion, M0, is the value of the variable that has the highest frequency. Bear in mind In some statistical studies there may be more than one trend. For data grouped in intervals, the most frequent interval is called the modal interval.

Mode

Position parameters

The scores obtained by Clara in the exercises carried out in the gymnastics championship were: 6, 7, 6, 6, 8, 9, 7. What is the fashion? The highest score is 6, as it appears 3 times Mo = 6.

example

The simple arithmetic mean, x , is the result of dividing the sum of all data by the total number of data. Bear in mind If the data is given in a frequency table, the mean is found by multiplying each value by its frequency and dividing by the total number of data. For data grouped in intervals the class marks of the intervals are taken as values of the xi.

Example

Position parameters

Clara’s average score is obtained by adding the 7 scores and dividing the result by 7.

Mean

To find the weighted average, add the products of each data by their weight and divide the result by the sum of the weights.

example

The median, M, is the central value of the variable, that is, the number of data that are less than it matches the number of data that are greater. To calculate the median, the data are ordered from lowest to highest: If the data number is odd, it is the data that occupies the center of the distribution. If the data number is even, it is the arithmetic mean of the two central values. Bear in mind Dividing the ordered data into four equal parts yields the quartiles: Q1, Q2 and Q3. For data grouped in intervals, the range in which the median is located is called the median range.

Median

Position parameters

To obtain the median of Clara’s scores, the data are ordered from lowest to highest: 6 6 6 (7) 7 8 9. Sorted data from minor to major. The central value is the median, M = 7. If Clara performs one more exercise with a score of 6, the median is: M= 6+7/2=6.5

Dispersion parameters

Sometimes it is necessary to know if the data is close to the center or far away.

example

The range is the difference between the largest and the smallest of the values taken by the variable.

range

Dispersion parameters

Calculate the temperature range of the two land areas. Route zone A: 24-12 = 12 Route area B: 20-17 = 3

Bear in mind If the data are grouped in intervals the path is the difference between the top end of the largest range and the bottom end of the smallest range.

example

In statistics the average absolute deviation or, simply mean or average deviation of a data set is the mean of the absolute deviations and is a summary of the statistical dispersion. According to this formula:

mean desviation

Dispersion parameters

Calculate the average temperature deviation of the two zones.

example

Another way to prevent negative and positive deviations from averaging is to use the parameters that use square deviations. The variance, , is the mean of the squares of the deviations of the data from the mean of the data.

variance

Dispersion parameters

Calculate the temperature variance of these two terrestrial zones:

When using the squares of the deviations, the variance and the variable will not have the same units, for example, if the variable is in kilometers, the variance will be given in square kilometers. To avoid this, the square root of the variance is used. The standard deviation, s, is the positive square root of the variance.

Standard desviation

Dispersion parameters

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