STATISTICS a Sup What is STATISTICS A set

STATISTICS © a. Sup

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. © a. Sup

Two General Purpose of Statistics (Gravetter, 2007) 1. 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 2. 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 © a. Sup

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

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 © a. Sup

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 © a. Sup

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 © a. Sup

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

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 © a. Sup

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 © a. Sup

TWO KINDS OF NUMERICAL DATA Generally fall into two major categories: 1. Counted frequencies enumeration data 2. Measured metric or scale values measurement or metric data Statistical procedures deal with both kinds of data © a. Sup

DATUM and DATA § The measurement or observation obtain for each individual is called a datum or, 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 © a. Sup

VARIABLE § A variable is a characteristic or condition that changes or has different values for different individual § A constant is 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 © a. Sup

QUALITATIVE and QUANTITATIVE Categories § 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. © a. Sup

DISCRETE and CONTINUOUS Variables § A discrete variable. No values can exist between two neighboring categories. § A continuous variable 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 © a. Sup

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. © a. Sup

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. © a. Sup

MEASUREMENTS and SCALES (Stevens, 1946) Ratio Interval Ordinal Nominal © a. Sup

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. © a. Sup

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. © a. Sup

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 © a. Sup

INTERVAL Scale § The top of this illustration shows three temperatures in degree Celsius: 00, 500, 1000. It is tempting to think of 1000 C 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 1000 C is twice as hot as one of 500 C or that a rise from 400 C to 480 C is a 20% increase © a. Sup

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. © a. Sup

RATIO Scale § One thing is certain: Scales …the kinds just mentioned HAVE ZERO POINT. © a. Sup

Confucius, 451 B. C What I hear, I forget What I see, I remember What I do, I understand © a. Sup

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 © a. Sup

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 © a. Sup

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) © a. Sup

Variabilitas Menggambarkan: ○ variasi ○ jangkauan ○ heterogenitas-homogenitas dari pengukuran suatu kelompok © a. Sup

Beberapa Pengukuran Variabilitas § Jangkauan /range (JT) § Interquartile Range (Q) dan Semiinterquartile range § Varians (S 2) § Simpang Baku/Standard Deviation (S) © a. Sup

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 © a. Sup 31

§ 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 60 th percentile Percentile Rank refers to a percentage Percentile refers to a score © a. Sup 32

Initstereng!! Aoccdrnig to a rscheearch at an Elingsh uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoatnt tihng is that frist and lsat ltteer is at the rghit pclae. The rset can be a toatl mses and you can sitll raed it wouthit porbelm. Tihs is bcuseae we do not raed ervey lteter by it slef but the wrod as a wlohe. © a. Sup 33

PROBABILITY © a. Sup

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 Inferential Statistics Probability © a. Sup SAMPLE

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 © a. Sup

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

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. © a. Sup

PROBABILITY and THE 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 © a. Sup

PROBABILITY and THE 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 © a. Sup

PROBABILITY and THE 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 © a. Sup

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 © a. Sup 0 1 2 Number of heads in 2 coin tosses

TOSSING COIN Number of heads in 3 coin tosses Number of heads in 4 coin tosses © a. Sup

The BINOMIAL EQUATION (p + © a. Sup n q)

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? © a. Sup

PROBABILITY and NORMAL DISTRIBUTION σ μ 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 © a. Sup

PROBABILITY and NORMAL DISTRIBUTION μ X Proportion below the curve B, C, and D area © a. Sup

B and C area X © a. Sup

B and C area X © a. Sup

B, C, and D area μ X B+C=1 C+D=½ B–D=½ © a. Sup

B, C, and D area X μ B+C=1 C+D=½ B–D=½ © a. Sup

The NORMAL DISTRIBUTION following a z-SCORE transformation 34. 13% 13. 59% 2. 28% -2 z -1 z 0 μ © a. Sup +1 z +2 z

34. 13% σ=7 13. 59% 2. 28% -2 z -1 z 0 μ = 166 +1 z +2 z Assume that the population of Indonesian adult height forms a normal shaped with a mean of μ = 166 cm and σ = 7 cm • p (X) > 180? • p (X) < 159? © a. Sup

34. 13% σ=7 13. 59% 2. 28% -2 z -1 z 0 +1 z μ = 166 +2 z Assume that the population of Indonesian adult height forms a normal shaped with a mean of μ = 166 cm and σ = 7 cm • Separates the highest 10%? • Separates the extreme 10% in the tail? © a. Sup

34. 13% σ=7 13. 59% 2. 28% -2 z -1 z 0 +1 z μ = 166 +2 z Assume that the population of Indonesian adult height forms a normal shaped with a mean of μ = 166 cm and σ = 7 cm • p (X) 160 - 170? • p (X) 170 - 175? © a. Sup

Chapter 8 INTRODUCTION TO HYPOTHESIS TESTING © a. Sup 56

The Logic of Hypothesis Testing § It usually is impossible or impractical for a researcher to observe every individual in a population § Therefore, researchers usually collect data from a sample and then use the sample data to answer question about the population § Hypothesis testing is statistical method that uses sample data to evaluate a hypothesis about the population © a. Sup 57

The Hypothesis Testing Procedure 1. State a hypothesis about population, usually 2. 3. 4. the hypothesis concerns the value of a population parameter Before we select a sample, we use hypothesis to predict the characteristics that the sample have. The sample should be similar to the population We obtain a sample from the population (sampling) We compare the obtain sample data with the prediction that was made from the hypothesis © a. Sup 58

PROCESS OF HYPOTHESIS TESTING § It assumed that the parameter μ is known for the population before treatment § The purpose of the experiment is to determine whether or not the treatment has an effect on the population mean Known population before treatment Unknown population after treatment TREATMENT μ = 30 © a. Sup μ=? 59

EXAMPLE § It is known from national health statistics that the mean weight for 2 -year-old children is μ = 26 pounds and σ = 4 pounds § The researcher’s plan is to obtain a sample of n = 16 newborn infants and give their parents detailed instruction for giving their children increased handling and stimulation § NOTICE that the population after treatment is unknown © a. Sup 60

STEP-1: State the Hypothesis § H 0 : μ = 26 (even with extra handling, the mean at 2 years is still 26 pounds) § H 1 : μ ≠ 26 (with extra handling, the mean at 2 years will be different from 26 pounds) § Example we use α =. 05 two tail © a. Sup 61

STEP-2: Set the Criteria for a Decision § Sample means that are likely to be obtained if H 0 is true; that is, sample means that are close to the null hypothesis § Sample means that are very unlikely to be obtained if H 0 is false; that is, sample means that are very different from the null hypothesis § The alpha level or the significant level is a probability value that is used to define the very unlikely sample outcomes if the null hypothesis is true © a. Sup 62

The location of the critical region boundaries for three different los -1. 96 -2. 58 -3. 30 © a. Sup α =. 05 α =. 01 α =. 001 1. 96 2. 58 3. 30 63

STEP-3: Collect Data and Compute Sample Statistics § After obtain the sample data, summarize the appropriate statistic σM = σ √n M-μ z= σ M © a. Sup NOTICE § That the top of the z-scores formula measures how much difference there is between the data and the hypothesis § The bottom of the formula measures standard distances that ought to exist between the sample mean and the population mean 64

STEP-4: Make a Decision § Whenever the sample data fall in the critical region then reject the null hypothesis § It’s indicate there is a big discrepancy between the sample and the null hypothesis (the sample is in the extreme tail of the distribution) © a. Sup 65

LEARNING CHECK HYPOTHESIS TEST WITH z § A standardized test that are normally distributed with μ = 65 and σ = 15. The researcher suspect that special training in reading skills will produce a change in scores for individuals in the population. A sample of n = 25 individual is selected, the average for this sample is M = 70. § Is there evidence that the training has an effect on test score? © a. Sup 66

FACTORS THAT INFLUENCE A HYPOTHESIS TEST M-μ z= σ M σM = © a. Sup σ √n § The size of difference between the sample mean and the original population mean § The variability of the scores, which is measured by either the standard deviation or the variance § The number of score in the sample 67

DIRECTIONAL (ONE-TAILED) HYPOTHESIS TESTS § Usually a researcher begin an experiment with a specific prediction about the direction of the treatment effect § For example, a special training program is expected to increase student performance § In this situation, it possible to state the statistical hypothesis in a manner that incorporates the directional prediction into the statement of H 0 and H 1 © a. Sup 68

LEARNING CHECK A psychologist has developed a standardized test for measuring the vocabulary skills of 4 -year -old children. The score on the test form a normal distribution with μ = 60 and σ = 10. A researcher would like to use this test to investigate the hypothesis that children who grow up as an only child develop vocabulary skills at a different rate than children in large family. A sample of n = 25 only children is obtained, and the mean test score for this sample is M = 63. © a. Sup 69