PSYC 331 STATISTICS FOR PSYCHOLOGISTS

Transcription

1 PSYC 331 STATISTICS FOR PSYCHOLOGISTS Session 1 BASIC CONCEPTS IN STATISTICS Lecturer: Dr. Paul Narh Doku, Dept of Psychology, UG Contact Information: pndoku@ug.edu.gh College of Education School of Continuing and Distance Education 2014/ /2017 godsonug.wordpress.com/blog

2 Session Overview This session introduces you to an review of some basic statistical concepts that will enable you to understand the concepts and principles in inferential statistics. By the end of this session learners should be able to clearly distinguish between descriptive and inferential statistics, have a good grasp of some summation notation, have a comprehensive understanding of the different scales of measurement used in social and educational research and their relation to statistical test and finally clearly understand what is meant by the mean, standard deviation and variance of a set of numbers or observations. Dr. P. N. Doku, Slide 2

3 Session Outline The key topics to be covered in the session are as follows: Topic one - Descriptive and inferential statistics Topic Two- Summation notation Topic Three - Measurement and scales of measurement in the Social Sciences and Education Topic Four - Scales of measurement and their applications to statistical tests Topic Five - The mean, standard deviation and variance of data measured on an interval or ratio scale Topic Six - Calculating the standard deviation and variance from raw scores Slide 3

4 Reading List Opoku, J. Y. (2007). Tutorials in Inferential Social Statistics. (2nd Ed.). Accra: Ghana Universities Press. Pages 3 22 MiĐhael Baƌƌoǁ, StatistiĐs foƌ EĐoŶoŵiĐs, AĐĐouŶtiŶg aŷd BusiŶess Studies, 4 th Edition, Pearson R.D. Mason, D.A. Lind, and W.G. Marchal, StatistiĐal TeĐhŶiƋues iŷ BusiŶess aŷd EĐoŶoŵiĐs, ϭϭ th Edition, McGraw- Hill Slide 4

5 TOPIC ONE: DESCRIPTIVE AND INFERENTIAL STSTISTICS WHAT IS STATISTIC? Statistic is a very broad subject, with applications in a vast number of different fields. Statistics has TWO meanings; namely, a discipline/subject and computation based on sample data (a sample calculation) 5

6 Descriptive and inferential statistics There are two major types of statistics (viewed as a sample calculation). Descriptive Statistics Inferential Statistics. The branch of statistic devoted to the summarization and description of data is called DESCRIPTIVE STATISTICS (data spread and central tendency) and the branch of statistics concerned with using sample data to make an inference about a population of data Slide 6 is called INFERENTIAL STATISTICS.

7 Key Terminologies: The concept of Population and Sample Population and sample are two basic concepts of statistics. A POPULATION may be defined as the totality/ collection of all objects, animals (including human beings), things, events, or even happenings. Example, all students following academic programmes at the University of Ghana make up the population of students at the University of Ghana. But often only a subset(small portion) of individuals of that population may be observed; such a subset of individuals constitutes a SAMPLE. Example, students offering Psychology in University of Ghana will constitute a sample for 7

8 The concept of sample A sample is a part of a population. In the example I gave above on the population of students at the University of Ghana, the total number of students studying under distance learning forms part of the total number of students following academic programmes at the University of Ghana and hence will be a sample based on the population of students at the University of Ghana.

9 Descriptive statistics Descriptive statistics are generally procedures for summarizing and describing quantitative information or data. In descriptive statistics Descriptive statistics includes the construction of graphs, charts, and tables, and the calculation of various descriptive measures such as averages, measures of variation, and percentiles. In descriptive statistics, We cannot tell exactly what is happening in the population from the description BUT can make guesses of what may be happening in the population because only part of the population is studied when we take a sample from a population.

10 Inferential Statistics Inferential statistics is solely concerned about techniques of drawing inferences (conclusions) from a sample to a population. As an example, suppose that you were interested in determining the average annual income of all residents in Accra, Ghana. Let us assume that there are ten million (10,000,000) households in Accra. The 10,000,000 households represent the population of households in Accra. It will obviously not only be expensive but also time-consuming to go to every household and determine the income levels of the occupants. Instead, you might obtain a list of all households in Accra and from this list randomly select about 500 households, visit the selected households and obtain the income levels of the occupants of each household. The 500 selected households represent a sample from the population of all households in Accra. From this data, you can calculate the mean (average) annual income and standard deviation based on the sample of 500 households.

11 Inferential Statistics cont. Let us assume that you find that the average annual income based on your sample is (GH ) with a standard deviation of (GH 30.00). From this data, you can then say that given a certain margin of error (which is very small if the sample was carefully selected from the population), then the population mean and standard deviation are also approximately equal to GH and GH respectively. Thus, you have made an inference or drawn a conclusion from a sample to a population. Note that this inference (estimate of population mean and standard deviation from the sample data) can be either right or wrong depending on how your sample was selected (only rich people or poor people ). It will be a biased sample and the sample mean and standard deviation will not be correct estimates of the population.

12 s 2 Sample statistics All computations made based on sample information is called a sample statistic. In the example above, the average annual income of GH and standard deviation of GH are estimates based on a sample. Usually denoted by ENGLISH ALPHABETS (e.g. for sample mean; s for sample standard deviation; S 2 for sample variance). You must remember then that anytime you come across English alphabets, it means that in most (but not in all) situations you are dealing with a sample.

13 Population parameter Estimations based on the entirety of the population becomes the population parameter. In the previous example, if you had studied the whole population of 10,000,000 households, then the average annual income and standard deviation that you calculate would be estimates based on a population. Usually symbolized by small GREEK ALPHABETS (e.g., for population mean; for population standard deviation; 2 for population variance). In effect, inferential statistics that this course is also about is concerned about techniques used in drawing inferences from sample statistics to their corresponding population parameters.

14 Dr. Richard Boateng, UGBS Slide 14

15 Dr. Richard Boateng, UGBS Slide 15

16 SUMMATION NOTATION Some notations that are frequently used when you are adding up scores or data collected for statistical analysis. These notations include Sum of a set of numbers or observations Sum of squares of a set of numbers or observations Square of the sum of a set of numbers or observations Sum of the product of two sets of numbers or observations Sum of squares of the product of two sets of numbers or observations Square of the sum of the product of two sets of numbers or observations Σ. Sigŵa ŵeaŷiŷg suŵŵatioŷ

17 Sum of a set of numbers or observations The capital Greek letter Σ, ;pƌoŷouŷđed sigŵa, staŷds foƌ suŵ of iŷ statistiđs. Suppose you ǁaŶt to fiŷd the suŵ of the following numbers: 2, 4, 6, 8, and 12. If you let represent any number, then you can refer to the five numbers listed above as and, where the subscripts 1, 2, 3, 4, and 5 stand for the ordinal positions of the numbers (the order in which the numbers are written).

18 Sum of a set of numbers or observations In the above example, and. In general, if there are n numbers in a set of measurements, then the sum of the n numbers can be symbolically written as This notation simply means that you add up all the X values, starting from the first value of X, to the last value of X,. In our example above, i ranges from 1 to 5 and therefore can simply be written as.

19 Sum of a set of numbers or observations When there is no confusion, the subscript i and the superscript n in the notation may be dropped and therefore the sum of the above set of numbers may simply be written as

20 Sum of squares of a set of n numbers or observations Sum of squares means that each X value (number) as in the previous example must first be squared and the resulting values added up. Symbolically, the sum of squares of n numbers is written as or simply as. In the previous example in the preceding slide involving the five numbers 2, 4, 6, 8, and 12, = = 264

21 Square of the sum of a set of n numbers or observations Square of the sum means that all the X values must first be added up and the resulting sum squared. Symbolically, the square of the sum of n numbers is written as or simply as. In the illustrations so far involving the five numbers, 2, 4, 6, 8, and 12, = = = 1,024

22 Sum of the product of two sets of numbers or observations (X and Y) In social science research, you may sometimes be interested in finding out the relationship between two variables. (A variable is any quantity that changes among individuals or over time, for example, grades obtained by students in an examination). OŶe ǀaƌiaďle Đould ďe PeƌfoƌŵaŶĐe iŷ MatheŵatiĐs aŷd the seđoŷd ǀaƌiaďle, PeƌfoƌŵaŶĐe iŷ StatistiĐs. You ŵay haǀe a prediction that students who are good in Mathematics are also good in Statistics and students who are poor in Mathematics are also poor in Statistics.

23 Sum of the product of two sets of numbers or observations (X and Y) Let us ƌepƌeseŷt oŷe ǀaƌiaďle, say PeƌfoƌŵaŶĐe iŷ MatheŵatiĐs ďy the syŵďol X aŷd let us ƌepƌeseŷt the otheƌ ǀaƌiaďle, PeƌfoƌŵaŶĐe iŷ StatistiĐs ďy the syŵďol Y. IŶ this example, let us assume that examination performance in both Mathematics and Statistics is marked over 15. For simplicity, let us assume that five students took the examination in both Mathematics and Statistics and that the X values (performance in Mathematics) were 2, 4, 6, 8, and 12, and that the Y values (performance in Statistics) for the same students were also 3, 4, 5, 7, and 10 respectively. Please note that for each X value there must be a corresponding Y value since each student was assessed on both the X and Y variables.

24 Sum of the product of two sets of numbers or observations (X and Y) The above data can be arranged in a table as: STUDENT NO. Score in Maths(X) Score in Stats(Y) XY Totals ΣXY=228

25 Sum of the product of two sets of numbers or observations (X and Y) Alternatively the sum of the product of the X and Y values could be symbolically written as ΣXY and thus solve accordingly as: = = 228 Slide 25

26 Sum of squares of the product of two sets of numbers or observations (X and Y) The sum of squares of the product of two sets of numbers or observations, X and Y, is symbolically written as. That is, we multiply the corresponding X and Y values, square each obtained product and add up the resulting products. In the example given so far: = = , ,400 = 18,728

27 Square of the sum of the product of two sets of observations (X and Y) The square of the sum of the product of two sets of numbers or observations (X and Y) is symbolically written as. That is, we first find the sum of the product of the two sets of numbers as in the previous example above and square this sum. Using the same figures in the previous examples = = = = 51,984

28 MEASUREMENT AND SCALES OF MEASUREMENT IN THE SOCIAL SCIENCES AND EDUCATION A measurement scale can possibly possess three properties: the properties of magnitude, equal intervals and an absolute zero point. Measurement in the Social Sciences and Education may range from physical and behavioural measures (e.g., height, weight, distance, etc., for physical measurements; and anxiety, depression, self- esteem, etc., for behavioural measurements); through rankings of physical and behavioural measures (e.g. 1st, 2nd, 3rd, etc.); to mere classification of objects, events, or happenings (e.g., male/female, present/absent, Yes/No, etc.).

29 Properties of Scales of Measurement Scales of measurement in social and behavioural research have three(3) basic properties: The property of Magnitude The property of Equal Intervals Magnitude--pƌopeƌty of ŵoƌeŷess. Higher score refers to more of something. Equal intervals--is the difference between any two adjacent numbers referring to the same amount of difference on the attribute? Absolute zero--does the scale have a zero point that refers to having none of that attribute? The property of Absolute Zero point

30 Property of Magnitude When a scale of measurement has magnitude, it means that an instance of whatever is being measured on the scale can be judged greater than ( > ), less than ( < ), or equal to ( = ) another instance of what is being measured. For example, 5 kilometres (5km) is greater than ( > ) a distance of 3 km; If the distance between two towns, A and B, is 10 km and the distance between two towns, C and D, is also 10km, then the distance between towns A and B is equal to ( = ) the distance between towns C and D. Items could be ranked, 1 st, 2 nd Slide, 3 rd 30 etc.

31 Property of Equal Interval When a measurement scale has equal intervals, it means that a unit of measurement between any two points on the scale is the same regardless of where the two points fall on the scale. For example, in the measurement of distance using the metre rule, the distance between 10cm and 11 cm is the same as the distance between 75cm and 76cm on the scale. Please note that any measurement scale that possesses the property of equal intervals also possesses the property of magnitude. In the measurement of distance, (11cm > 10cm), Slide 31

32 Property of Absolute Zero Point When we say that a measurement scale has an absolute zero point, it means that (in theory), at one end of the scale, nothing at all of whatever is being measured exists. For example, we may think of a distance of 0mm (no distance) or a weight of 0mg (a weightless object) if our units of measurement for distance and weight are millimetres and milligrams respectively. We may design a scale to measure something where we may decide to place zero (0) at one end of the scale. However, this does not necessarily mean that at that end of the scale, nothing at all of what we are measuring exists. Slide 32

33 Property of Absolute Zero Point cont: For example, one thermometer that we use to measure temperature starts from 0 o C to 100 o C on the Celsius scale. In more technical language, this thermometer is calibrated from 0 o C to 100 o C. Does it mean that when the thermometer reading is 0oC today, it means that there is no temperature? Absolutely no! The point 0 o C is merely chosen for convenience as the starting point of the scale to represent the melting point of ice. An important characteristic of a measurement scale with an absolute zero point is that we can make ratio statements on measurements with the scale. For example, if a boy weighs 30kg and a man weighs 60kg, we can say that the man weighs twice as the ďoy; oƌ the ďoy s ǁeight is half the ŵ Slid a e 3 Ŷ 3 s

34 Types of measurement scales

35 Nominal Scale Assigns a value to an object for identification or classification purposes. Most elementary level of measurement. Nominal is hardly measurement. It refers to quality more than quantity. A nominal level of measurement is simply a matter of distinguishing by name, e.g., 1 = male, 2 = female. Even though we are using the numbers 1 and 2, they do not denote quantity.

36 Nominal Scale cont: They are categories or classifications, examples: MEAL PREFERENCE: Breakfast, Lunch, Dinner RELIGIOUS PREFERENCE: 1 = Buddhist, 2 = Muslim, 3 = Christian, 4 = Jewish, 5 = Other POLITICAL ORIENTATION: Republican, Democratic, Libertarian, Green

37 Ordinal Scale Ordinal refers to order in measurement. Ranking scales allowing things to be arranged based on how much of some concept they possible. Have nominal properties An ordinal scale indicates direction, in addition to providing nominal information. Low/Medium/High; or Faster/Slower are examples of ordinal levels of measurement. Ranking an experience as a "nine" on a scale of 1 to 10 tells us that it was higher than an experience ranked as a "six." Many psychological scales or inventories are at the ordinal level of measurement.

38 Ordinal Scale cont: Examples: RANK: 1st place, 2nd place,... last place LEVEL OF AGREEMENT: No, Maybe, Yes POLITICAL ORIENTATION: Left, Center, Right

39 Interval Scale Capture information about differences in quantities of a concept. Have both nominal and ordinal properties. Interval scales provide information about order, and also possess equal intervals. From the previous example, if we knew that the distance between 1 and 2 was the same as that between 7 and 8 on our 10-point rating scale, then we would have an interval scale.

40 Interval Scale cont: Examples: TIME OF DAY on a 12-hour clock POLITICAL ORIENTATION: Score on standardized scale of political orientation OTHER scales constructed so as to possess equal intervals Interval time of day - equal intervals; analog (12-hr.) clock, difference between 1 and 2 pm is same as difference between 11 and 12 am

41 Ratio Scale Highest form of measurement. Have all the properties of interval scales with the additional attribute of representing absolute quantities. Absolute zero - In addition to possessing the qualities of nominal, ordinal, and interval scales, a ratio scale has an absolute zero (a point where none of the quality being measured exists). Using a ratio scale permits comparisons such as being twice as high, or one-half as much.

42 Ratio Scale cont: Examples: RULER: inches or centimeters YEARS of work experience INCOME: money earned last year NUMBER of children GPA: grade point average

43 Scales of Measurement and Behavioural Measurements As you can see from the examples given, scales with absolute zero points are mainly to be found in the Physical Sciences. Most of the scales of measurement used in the Social Sciences and Education possess only the property of magnitude and in some cases, equal intervals. Take the measurement of a human trait or characteristic like aggression as an example. In the measurement of aggression, it does not make sense to think of a scale where nothing at all of the aggressive trait or characteristic exists at one end of the scale. This is because every human being has some level of aggression within him/her. It is a matter of degree to which individuals show this aggression. Thus, any scale designed to measure aggression, depression, anxiety, attitude toward some object, etc., will have no absolute zero point. At best, such a scale may have equal intervals, which in most cases are assumed to be the case when in fact it may not really be so.

44 SCALES OF MEASUREMENT AND THEIR RELATIONSHIPS TO STATISTICAL The TESTS choice of a statistical test to analyze data will depend, to some extent, on the measurement scale used to collect the data. Thee are 2 broad groups of tests to analyze data. Parametric statistical tests that make assumptions about the shape of the distribution of data in the population are employed when the level of measurement is either ratio or interval. Nonparametric or distribution-free statistical tests are statistical tests that do not make any assumptions about the shape of the distribution of data in the population. They are used when the level of measurement is either ordinal or nominal.

45 Parametric statistical tests Examples The z test t tests one sample t test, independent t test, dependent or correlated t test The ANOVA or F test, and The Pearson product- moment correlation Assumptions Ratio/Interval Data collected on the outcome/dependent variable Normality: Data have a normal distribution (or at least is symmetric) Homogeneity of variances: Data from multiple groups have the same variance. Linearity: Data have a linear relationship.

46 Non-Parametric statistical test Examples Spearman rank-order correlation coefficient (abbreviated as Spearman ) Chi square test. Mann- Whitney U test Wilcoxon matched-pairs signed- ranks test, Spearman Rho Kruskal-Wallis H test. Slide 46

47 Types of Statistical tests that are used with various scales of measurement Scale of Types of Test to use Examples of Specific Tests to use Measurement Ratio Interval Parametric tests measurement DIFFERENCES: Z test, one sample t test, independent t test, dependent or correlated t test, One way ANOVA (F) test, RELATIONSHIP: Pearson r Ordinal Nominal Nonparametric Tests DIFFERENCES: Wilcoxon matched-pairs signed- ranks test, Kruskal-Wallis H test. Mann-Whitney U test, RELATIONSHIPS Spearman Chi Square test Slide 47

48 THE MEAN, STANDARD DEVIATION AND VARIANCE OF DATA MEASURED ON AN INTERVAL OR RATIO SCALE: THE MEAN In inferential statistics, the mean refers to the arithmetic mean. It is perhaps the most widelyused measure of central tendency The arithmetic mean of a set of numbers is simply the sum of the numbers

49 Mean: For example, to find the mean of the following numbers: 5, 6, 8, 10, 15, and 16, you simply find the sum of the numbers and divide this sum by the total number of observations, which is equal to six (6) in this case. Thus, the mean of the above set of numbers = = Slide 49

50 Mean: In general, if we let Xi or simply X stand for any number in a set of numbers, then the sum of the numbers in the set = ΣX. If we let N or Ŷ stand for the total number of observations, then the mean can generally be written as N is normally used to stand for population size while Ŷ is used to stand for sample size

51 Mean The mean based on POPULATON data is symbolized by Population mean: the mean based on a sample symbolized by Sample mean: Slide 51

52 Measurement of variability of a set of numbers or observations Variability refers to the extent to which a set of numbers or observations vary about or deviate from a measure of central tendency (the mean). Measures of variability are also sometimes called measures of dispersion. Examples include the standard deviation, the variance, and the range

53 Measurement of variability of a set of numbers or observations As an illustration, consider the following two sets of numbers: (i) 10, 11, 11, 12, 13, 12, 10, 13; and (ii) 10, 14, 19, 23, 30, 29, 45, 50. If you carefully inspect the two sets of data, you will notice that the numbers in set number (i) are close to each other in values than the numbers in set number (ii). This means that the numbers in set (ii) show greater variability than the numbers in set (i).

54 Measurement of variability of a set of numbers or observations Now, consider a situation where the numbers 5,6,8,10,15,16 stand for the scores (out of 20) obtained by six students in an examination in Mathematics with a mean score of 10. If the six students were of equal ability in all respects, then it is reasonable to expect that each student would have obtained the score of 10, the mean score But it is totally unrealistic to expect that all the six students would be of equal ability due to individual differences on factors like intelligence, interest, motivation, practice, etc. For this reason, we expect that some students will obtain higher scores than others The reason for calculating variability is to determine the extent to which the scores of these six students in our example, vary about or deviate from the mean score of 10.

55 A practical demonstration of the concept of variability To find this variability or deviation, you may think that you can do this by siŵply suďtƌađtiŷg eađh studeŷt s sđoƌe fƌoŵ the ŵeaŷ sđoƌe of ϭϭ as folloǁs: (5-10); (6-10); (8-10); (10-10); (15-10); and (16-10). You may then add up the values to obtain a measure of variability, i.e., variability = (-5) + (- 4) + (-2) + (0) + (5) + (6) = (-11) + (11) = 0. You see what has happened? The figure 0 (zero) that you obtained means that there is no variability. In other words, the students are all obtaining the same score! However, by inspecting the data, we notice differences in individual scores, hence there is variability. The reason why we are obtaining zero here is that in Mathematics, for any set of numbers, the sum of the difference between the numbers and their mean is always equal to zero. [You may try proving this with another set of numbers]

56 The concept of variance A way out of the difficulty we are faced with in calculating variability using the simple logic in preceding slide will be for us to use the squares of the numbers, instead of the raw numbers, In other words, the squares of the difference between the set of scores (numbers) and their mean are used to calculate variability. Using this method, the variability of the scores recorded for the six students in our example will be: (5-10) 2 + (6-10) 2 + (8-10) 2 + (10-10) 2 + (15-10) 2 + (16-10) 2 = (-5) 2 + (-4) 2 + (-2) 2 + (0) 2 + (5) 2 + (6) 2 = = 106. The computed value of 106 is the total variability of the 6 scores about their mean Variance is the average variability which is 106/6» (corrected to 2 decimal places)

57 Formula for variance Population variance symbolized by Sample variance symbolized by Slide 57

58 The concept of standard deviation Note that because deviation of each score from the mean was squared to estimate the variance in the previous example, variance is in square units whereas the original measurements are not in square units. The standard deviation is simply the square root of the variance and is preferred over the variance because: 1. Taking the square root of the variance (i.e., finding the standard deviation) brings the measure of variability or dispersion to the same unit of measurement as the original measurement of the ten individuals. For example supposing the measurement were in (cm), variance will square them and hence (cm 2 ); finding the square root(standard deviation brings the unit back to cm(original unit. 2. The value of the standard deviation is normally less then the mean and being measured in the same units as the mean, it becomes possible to roughly estimate the spread of scores about the mean

59 Formula for standard deviation Population parameter Sample statistic Slide 59

60 CALCULATING THE STANDARD DEVIATION AND VARIANCE FORM A RAW DATA It is relatively easier to calculate the standard deviation and variance from raw scores than calculating these values using the deviations from the mean method just illustrated, but that the different methods in calculating will give you the same values.

61 The standard deviation and variance based on a population Fortunately, there is a relatively easier way of calculating the standard deviation and variance based on the raw scores without going through the tedious exercise involved in the calculations using the previous formulae (deviation from the mean). This easier formula which may be used for any population and sample sizes is based on sum of squares and squares of the sum of the raw

62 The standard deviation and variance based on a population Population standard deviation To get the variance, square the standard deviation. Please note again that in the above stand formula, for sum of squares of the raw data stands for while the square of the sum of the raw scores.

63 The standard deviation and variance based on a sample The standard deviation and variance based on a sample is also calculated from the raw data using sum of squares and square of the sum of the data using the following formula: IŶ the aďoǀe foƌŵula, s as usual stands for the standard deviation and n stands for the sample size. Note that ǁe ĐalĐulate the saŵple statistiđ, s, to estiŵate the corresponding population parameter,.

64 References Opoku, J. Y. (2007). Tutorials in Inferential Social Statistics. (2nd Ed.). Accra: Ghana Universities Press. Pages 3 22 MiĐhael Baƌƌoǁ, StatistiĐs foƌ EĐoŶoŵiĐs, AĐĐouŶtiŶg aŷd BusiŶess Studies, 4 th Edition, Pearson R.D. MasoŶ, D.A. LiŶd, aŷd W.G. MaƌĐhal, StatistiĐal TeĐhŶiƋues iŷ BusiŶess aŷd EĐoŶoŵiĐs, ϭϭ th Edition, McGraw-Hill Slide 64