STATISTICS

1. STATISTICS

2. What is STATISTICS? • A set of mathematical procedure for organizing, summarizing, and interpreting information (Gravetter, 2004) • A branch of mathematics which specializes in enumeration data and their relation to metric data (Guilford, 1978) • Any numerical summary measure based on data from a sample; contrasts with a parameter which is based on data from a population (Fortune, 1999) • etc.

3. Two General Purpose of Statistics (Gravetter, 2007) • Statistic are used to organize and summarize the information so that the researcher can see what happened in the research study and can communicate the result to others • Statistics help the researcher to answer the general question that initiated the research by determining exactly what conclusions are justified base on the result that were obtained

4. DESCRIPTIVE STATISTICS The purpose of descriptive statistics is to organize and to summarize observations so that they are easier to comprehend

5. INFERENTIAL STATISTICS The purpose of inferential statistics is to draw an inference about condition that exist in the population (the complete set of observation) from study of a sample (a subset) drawn from population

6. SOME TIPS ON STUDYING STATISTICS • Is statistics a hard subject? IT IS and IT ISN’T • In general, learning how-to-do-it requires attention, care, and arithmetic accuracy, but it is not particularly difficult. LEARNING THE ‘WHY’ OF THINGS MAY BE HARDER

7. SOME TIPS ON STUDYING STATISTICS • Some parts will go faster, but others will require concentration and several readings • Work enough of questions and problems to feel comfortable • What you learn in earlier stages becomes the foundation for what follows • Try always to relate the statistical tools to real problems

8. POPULATIONS and SAMPLES THE POPULATION is the set of all the individuals of interest in particular study The sample is selected from the population The result from the sample are generalized from the population THE SAMPLE is a set of individuals selected from a population, usually intended to represent the population in a research study

9. PARAMETER and STATISTIC • A parameter is a value, usually a numerical value, that describes a population. A parameter may be obtained from a single measurement, or it may be derived from a set of measurements from the population • A statistic is a value, usually a numerical value, that describes a sample. A statistic may be obtained from a single measurement, or it may be derived from a set of measurement from sample

10. SAMPLING ERROR • It usually not possible to measure everyone in the population • A sample is selected to represent the population. By analyzing the result from the sample, we hope to make general statement about the population • Although samples are generally representative of their population, a sample is not expected to give a perfectly accurate picture of the whole population • There usually is some discrepancy between sample statistic and the corresponding population parameter called sampling error

11. TWO KINDS OF NUMERICAL DATA Generally fall into two major categories: • Counted frequencies  enumeration data • Measured  metric or scale values  measurement or metric data Statistical procedures deal with both kinds of data

12. DATUM and DATA • The measurement or observation obtain for each individual is called a datumor, more commonly a score or raw score • The complete set of score or measurement is called the data set or simply the data • After data are obtained, statistical methods are used to organize and interpret the data

13. VARIABLE • A variable is a characteristic or condition that changes or has different values for different individual • A constantis a characteristic or condition that does not vary but is the same for every individual • A research study comparing vocabulary skills for 12-year-old boys

14. QUALITATIVE and QUANTITATIVECategories • Qualitative: the classes of objects are different in kind. There is no reason for saying that one is greater or less, higher or lower, better or worse than another. • Quantitative: the groups can be ordered according to quantity or amount It may be the cases vary continuously along a continuum which we recognized.

15. DISCRETE and CONTINUOUS Variables • A discrete variable. No values can exist between two neighboring categories. • A continuousvariable is divisible into an infinite number or fractional parts • It should be very rare to obtain identical measurements for two different individual • Each measurement category is actually an interval that must be define by boundaries called real limits

16. CONTINUOUS Variables • Most interval-scale measurement are taken to the nearest unit (foot, inch, cm, mm) depending upon the fineness of the measuring instrument and the accuracy we demand for the purposes at hand. • And so it is with most psychological and educational measurement. A score of 48 means from 47.5 to 48.5 • We assume that a score is never a point on the scale, but occupies an interval from a half unit below to a half unit above the given number.

17. FREQUENCIES, PERCENTAGES, PROPORTIONS, and RATIOS • Frequency defined as the number of objects or event in category. • Percentages (P) defined as the number of objects or event in category divided by 100. • Proportions (p). Whereas with percentage the base 100, with proportions the base or total is 1.0 • Ratio is a fraction. The ratio of a to b is the fraction a/b. A proportion is a special ratio, the ratio of a part to a total.

18. MEASUREMENTS and SCALES (Stevens, 1946) Ratio Interval Ordinal Nominal

19. NOMINAL Scale • Some variables are qualitative in their nature rather than quantitative. For example, the two categories of biological sex are male and female. Eye color, types of hair, and party of political affiliation are other examples of qualitative or categorical variables. • The most limited type of measurement is the distinction of classes or categories (classification). • Each group can be assigned a number to act as distinguishing label, thus taking advantage of the property of identity. • Statistically, we may count the number of cases in each class, which give us frequencies.

20. ORDINAL Scale • Corresponds to was earlier called “quantitative classification”. The classes are ordered on some continuum, and it can be said that one class is higher than another on some defined variable. • All we have is information about serial arrangement. • We are not liberty to operate with these numbers by way of addition or subtraction, and so on.

21. INTERVAL Scale • This scale has all the properties of ordinal scale, but with further refinement that a given interval (distance) between scores has the same meaning anywhere on the scale. Equality of unit is the requirement for an interval scales. • Examples of this type of scale are degrees of temperature. A 100 in a reading on the Celsius scale represents the same changes in heat when going from 150 to 250 as when going from 400 to 500

22. INTERVAL Scale • The top of this illustration shows three temperatures in degree Celsius: 00, 500, 1000. It is tempting to think of 1000C as twice as hot as 500. • The value of zero on interval scale is simply an arbitrary reference point (the freezing point of water) and does not imply an absence of heat. • Therefore, it is not meaningful to assert that a temperature of 1000C is twice as hot as one of 500C or that a rise from 400C to 480C is a 20% increase

23. INTERVAL Scale • Some scales in behavioral science are measurement of physical variables, such as temperature, time, or pressure. • However, one must ask whether the variation in the psychological phenomenon is being measured indirectly is being scaled with equal units. • Most measurements in the behavioral sciences cannot posses the advantages of physical scales.

24. RATIO Scale • One thing is certain: Scales …the kinds just mentioned HAVE ZERO POINT.

25. Confucius, 451 B.C What I hear, I forget What I see, I remember What I do, I understand

26. Jenis-jenis statistika deskriptif yang telah dipelajari • Distribusi frekuensi: Menunjukkan seluruh skor yang ada dan frekuensi kemunculannya (ungrouped & grouped data) • Kurva Normal: Distribusi Normal dan Probabilitas, Proporsi, dan z-scores

27. Jenis-jenis statistika deskriptif yang telah dipelajari Tendensi sentral: To find the single score that is most typical or most representative of the entire group (Gravetter & Wallnau, 2007)  Mean, Median, Mode

28. Jenis-jenis statistika deskriptif yang telah dipelajari • Variabilitas: Measures the dispersion among the scores (or how spread out the data are) around the central measure (Furlong, 2000) • Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together (Gravetter & Wallnau, 2007)

29. Variabilitas Menggambarkan: • variasi • jangkauan • heterogenitas-homogenitas dari pengukuran suatu kelompok

30. Beberapa Pengukuran Variabilitas • Jangkauan /range (JT) • Interquartile Range (Q) dan Semi-interquartile range • Varians (S2) • Simpang Baku/Standard Deviation (S)

31. PERCENTILES and PERCENTILE RANKS • The percentile system is widely used in educational measurement to report the standing of an individual relative performance of known group. It is based on cumulative percentage distribution. • A percentile is a point on the measurement scale below which specified percentage of the cases in the distribution falls • The rank or percentile rank of a particular score is defined as the percentage of individuals in the distribution with scores at or below the particular value • When a score is identified by its percentile rank, the score called percentile

32. Percentile Rank refers to a percentagePercentile refers to a score • Suppose, for example that A have a score of X=78 on an exam and we know exactly 60% of the class had score of 78 or lower….… • Then A score X=78 has a percentile of 60%, and A score would be called the 60th percentile

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34. PROBABILITY

35. INTRODUCTION TO PROBABILITY We introduce the idea that research studies begin with a general question about an entire population, but actual research is conducted using a sample POPULATION SAMPLE Inferential Statistics Probability

36. THE ROLE OF PROBABILITY IN INFERENTIAL STATISTICS • Probability is used to predict what kind of samples are likely to obtained from a population • Thus, probability establishes a connection between samples and populations • Inferential statistics rely on this connection when they use sample data as the basis for making conclusion about population

37. PROBABILITY DEFINITION The probability is defined as a fraction or a proportion of all the possible outcome divide by total number of possible outcomes Number of outcome classified as A Probability of A = Total number of possible outcomes

38. EXAMPLE • If you are selecting a card from a complete deck, there is 52 possible outcomes • The probability of selecting the king of heart? • The probability of selecting an ace? • The probability of selecting red spade? • Tossing dice(s), coin(s) etc.

39. PROBABILITY andTHE BINOMIAL DISTRIBUTION When a variable is measured on a scale consisting of exactly two categories, the resulting data are called binomial (two names), referring to the two categories on the measurement

40. PROBABILITY andTHE BINOMIAL DISTRIBUTION • In binomial situations, the researcher often knows the probabilities associated with each of the two categories • With a balanced coin, for example p (head) = p (tails) = ½

41. PROBABILITY andTHE BINOMIAL DISTRIBUTION • The question of interest is the number of times each category occurs in a series of trials or in a sample individual. • For example: • What is the probability of obtaining 15 head in 20 tosses of a balanced coin? • What is the probability of obtaining more than 40 introverts in a sampling of 50 college freshmen

42. TOSSING COIN • Number of heads obtained in 2 tosses a coin • p = p (heads) = ½ • p = p (tails) = ½ • We are looking at a sample of n = 2 tosses, and the variable of interest is X = the number of head The binomial distribution showing the probability for the number of heads in 2 coin tosses 0 1 2 Number of heads in 2 coin tosses

43. TOSSING COIN Number of heads in 3 coin tosses Number of heads in 4 coin tosses

44. The BINOMIAL EQUATION (p + q)n

45. LEARNING CHECK • In an examination of 5 true-false problems, what is the probability to answer correct at least 4 items? • In an examination of 5 multiple choices problems with 4 options, what is the probability to answer correct at least 2 items?

46. PROBABILITY and NORMAL DISTRIBUTION In simpler terms, the normal distribution is symmetrical with a single mode in the middle. The frequency tapers off as you move farther from the middle in either direction σ μ

47. PROBABILITY and NORMAL DISTRIBUTION Proportion below the curve  B, C, and D area μ X

48. B and C area X

49. B and C area X

50. B, C, and D area B + C = 1 C + D = ½  B – D = ½ μ X