1. ELEMENTS OF MATHEMATICAL STATISTICS

2. GENERAL NOTIONS Mathematical statistics is a set of methods that are used to collect, analyze group and interprete data concerning a certain phenomenon and predict its future evolution. THE STATISTICAL POPULATION is a definite set of similar objects. Elements of a statistical population are called statistical units or individuals. The part of the entire population for which the statistical analysis is made is called sample. CHARACTERISTIC ( or variable ) is the common feature of statistical population which we study in the statistical analysis. Characteristics are : - quantitative – if we can measure them (average, height) - qualitative – if we can`t measure them ( eyes color).

3. ELEMENTS OF MATHEMATICAL STATISTICS THE QUANTITATIVE CHARACTERISTICS can be : - discontinuous – if the variable takes a finite number of values - continuous – if the variable takes any value from a finite or infinite interval For example : the average mean in mathematics takes values: 1, 2, …, 10 and is a discontinuous characteristic and height can take any value in a finite interval and is a continuous characteristic. STAGES OF A STATISTICAL STUDY (survey) - collecting data - presenting data in a table from which expresses answers, allows comparing and classifying - representing data - calculating specific parameters - interpreting data

4. ELEMENTS OF MATHEMATICAL STATISTICS TOTAL EFFECTIVE is the total number of the units of a statistical population. ABSOLUTE FREQUENCY of a value of the characteristic is the number of units corresponding to this value RELATIVE FREQUENCY of a value is the fraction between the absolute frequency and the total effective. CUMULATIVE FREQUENCY ( increasing ) of a value of the characteristic is the sum of all relative frequencies of values up to and including this value

5. ELEMENTS OF MATHEMATICAL STATISTICS EXEMPLES GRAPHIC REPRESENTATION OF DATA Representation of statistical series with discontinuous quantitative variable: - by columns (on the horizontal axis – values of the variable on the vertical axis - values of absolute frequency ). Representation of statistical series with continuous quantitative variable: - by histogram ( on the horizontal axis- succession of segments proportional to the length of classes; there are built rectangles having these segments as bases and height proportional to the absolute frequencies ). 123

6. ELEMENTS OF MATHEMATICAL STATISTICS Representation of statistical series with qualitative variable : - by structural rectangle ( it is divided into rectangles of areas proportional to relative frequencies ), - by structural circle ( it is divided into circle sectors with areas proportional to relative frequencies ), - diagrams with columns or bands ( rectangles with equal bases and heights proportional to the absolute frequencies, placed at equal distances from each other ). 4657 EXEMPLES

7. ELEMENTS OF MATHEMATICAL STATISTICS EXEMPLU CHARACTERISTIC ELEMENTS OF A STATISTICAL SERIES The tables with the values of the characteristic and the corresponding absolute frequencies define the distribution of the statistical repartition of the statistical variable and represent a statistical series with only one variable. THE AVERAGE Considering the statistical variable x taking the values x 1, x 2, x 3, … x k with the corresponding effectives n 1, n 2, n 3, … n k Definition We call the average of the characteristics the number _ n 1 x 1 +n 2 x 2 +…+n k x k x= ——————————— n 1 +n 2 +…+n k

8. ELEMENTS OF MATHEMATICAL STATISTICS ~ Definition THE MEDIAN x, is a value for which half of the values x i are smaller or equal and the other half of the values x i are bigger or equal to that. Definition THE MODULUS ( or THE DOMINANT ) of a statistical series is the value of the characteristic corresponding to the highest frequency, in case of discontinuous values and the main value of the class corresponding to the highest frequency in case of continuous values.

9. Marks in Mathematics TABLE BACK

10. Marks in Mathematics MarksStudents Absolute frequency(f a ) Relative frequency(f r ) Cumulative frquency(f c ) 410.020.02 580.160.18 6 101010100.200.38 7 101010100.200.58 890.180.76 990.180.94 1030.061.00 CHART BACK

11. Height TABLE BACK

12. Height CHART BACK HeightStudents Absolute frequency(f a ) Relative frequency(f r ) Cumulative frquency(f c ) 410.020.02 580.160.18 6 101010100.200.38 7 101010100.200.58 890.180.76 990.180.94 1030.061.00

13. Distance from home to school TABLE BACK

14. Distance from home to school CHART BACK DistanceStudents Absolute frequency(f a ) Relative frequency(f r ) Cumulative frquency(f c ) 0-30060.120.12 300-60090.180.30 600-900 101010100.200.50 900-120090.180.68 1200-150040.080.76 1500-180070.140.90 1800-2100 50.101.00

15. Eye colour TABLE BACK Blue Green Brown Black 10% 28% 52% 10%

16. Eye colour Eye colour Students Blue5 Green14 Brown26 Black5 CHART BACK

17. Favourite sport TABLE BACK

18. Favourite sport SportStudents Voleyball15 Handball3 Basketball4 Football15 Tennis8 Other6 CHART BACK

19. Favourite music TABLE BACK

20. Favorite music TypeStudents House 11111111 Dance12 Rock6 Symphonic4 Heavy metal 3 Rap7 Other7 CHART BACK

21. Birth month TABLE BACK

22. Birth month MonthStudents January3 February2 March7 April4 May5 June4 July4 August2 September4 October6 November4 December5 CHART BACK

23. Survey carried out by students and staff of National College “August Treboniu Laurian” Botosani Romania