ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ

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1 ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ ΕΛΕΝΑ ΦΛΟΚΑ Επίκουρος Καθηγήτρια Τµήµα Φυσικής, Τοµέας Φυσικής Περιβάλλοντος- Μετεωρολογίας

2 ΓΕΝΙΚΟΙ ΟΡΙΣΜΟΙ Πληθυσµός Σύνολο ατόµων ή αντικειµένων στα οποία αναφέρονται οι παρατηρήσεις είγµα Υποσύνολο του πληθυσµού που επιλέχτηκε τυχαία Μεταβλητή Το χαρακτηριστικό το οποίο παρατηρούµε στον πληθυσµό ή το δείγµα

3 ΓΕΝΙΚΟΙ ΟΡΙΣΜΟΙ Ποσοτική µεταβλητή Η µεταβλητή που επιδέχεται µέτρηση µε πραγµατικές τιµές Ποιοτική µεταβλητή Η µεταβλητή που δεν επιδέχεται µέτρηση και εκφράζεται µε λέξεις Ποσοτικές µεταβλητές Συνεχείς Ασυνεχείς

4 ΓΕΝΙΚΟΙ ΟΡΙΣΜΟΙ Συχνότητα (ν i ) Πόσες φορές µια τιµή xi της µεταβλητής Χ παρουσιάζεται στο δείγµα Μέγεθος του δείγµατος (N) Σχετική Συχνότητα (f i ) fi = v v i 100 % k = = i 1 vi N Αθροιστική συχνότητα (Fi) (δεξιόστροφη) Το άθροισµα των συχνοτήτων v i των τιµών που είναι µικρότερες ή ίσες της τιµής x i Σχετική Αθροιστική συχνότητα (Fi) Το άθροισµα των σχετικών συχνοτήτων f i των τιµών που είναι µικρότερες ή ίσες της τιµής x i k fi = = 100 i 1 %

5 Precision Precision is an indication of how close the elements of a series of measurements are to each other. It is desirable to have results that are precise, the results are grouped close together and not scattered.

6 Precision When the measurements are characterised by small random errors (e.g. errors derived from experimental conditions resulting in different values during subsequent measurements of the same parameter)

7 Precision Good Precision

8 Precision Poor Precision

9 Accuracy Accuracy is a measure of the difference between a measured value and the true value, that is, the error. If the errors of measurement average to zero, then the system is said to be accurate. It is desirable to have a system that is accurate, that is, has results that are close to the true value.

10 Accuracy When the measurements are characterised by small systematic errors (e.g. standard biases of the measurements derived from instrument calibration, personal errors, experimental design)

11 Accuracy Good Accuracy

12 Accuracy Poor Accuracy

13 Precision and Accuracy A system may be precise even though it is not accurate. Conversely, a system may be accurate, but not precise.

14 Precision and Accuracy Good Precision Good Accuracy

15 Precision and Accuracy Possible Results Good Accuracy Poor Accuracy Good Precision Precise and Accurate Precise but Inaccurate Poor Precision Imprecise but Accurate Imprecise and Inaccurate

16 CLASSIFICATION AND TABULATION Raw data: not organised data Grouped data: data organised and summarised in classes Class interval: the interval chosen for classifying Class frequency: number of observations within a particular class Class limits: the end numbers of the class interval (lower and upper limits) Open class interval: a class interval that has either no upper class or lower class limit

17 CLASSIFICATION AND TABULATION Class boundaries: (upper limit of one class interval + lower limit of the next higher class interval)/2 e.g , class boundary=(62+63)/2= (lower boundary)-62.5(upper boundary) Class width or class size: difference between the lower and upper class boundaries e.g. ( )=3 Class mark or class midpoint: lower+upper class limits)/2 e.g. (60+62)/2=61

18 CLASSIFICATION AND TABULATION Example Class limits Class intervals Class midpoint (60+62)/2 =61 Class boundaries Frequ ency Relative frequenc y(%) x100/22 = (62+63)/ (65+66)/2= Class size= =3 N=22

19 Classification and tabulation General rules for classification Determine the range of the raw data (max-min) min) Find the number of class intervals A=5logN N=frequency of raw data Find class size range A

20 Classification and tabulation It is suggested that: Intervals have same size the number of intervals is usually 5-20 Midpoints usually coincide with observed data Class boundaries do not coincide with observations The distribution of the data within the classes should be as uniform as possible Closed classes are preferred

21 Classification and tabulation Example: N=28 max=76 min=63.2 A=5logN=5log28=7.2 number of class intervals= 7 Range=max-min= =12.8 Class size=12.8/7 2 1st interval: nd interval:

22 . CLASSIFICATION AND TABULATION Example: N=28 max=76 min=63.2 A=5logN=5log28=7.2 number of class intervals= 7 Range=max-min= =12.8 Class size=12.8/7 2 1 st interval: nd interval:

23 REPRESENTATION OF FREQUENCY DISTRIBUTIONS Relative frequency (%) Class intervals Frequency histogram and frequency polygon (or ogive)

24 REPRESENTATION OF FREQUENCY DISTRIBUTIONS Relative frequency (%) Class intervals Frequency curves

25 ) CUMULATIVE FREQUENCY Αθροιστική συχν ότητα (% ) Αθροιστική συχνότητα (% Κλίµακα Beaufort Κλίµακα Beaufort RF (%) CRF (%) RF (%) CRF (%) Less than cumulative frequency More than cumulative frequency

26 CALCULATION OF FREQUENCY WITH THE AID OF EXCEL Upper class limit

27 CALCULATION OF CUMULATIVE FREQUENCY WITH THE AID OF EXCEL

28 CALCULATION OF FREQUENCY WITH THE AID OF EXCEL Ctrl+Shift+Enter

29 CALCULATION OF FREQUENCY WITH THE AID OF EXCEL