Statistics Refresher PDF

Here is the transcription of the images into markdown format: ## A STATISTICS REFRESHER Prepared by: Jan Mari B. Mangalili, RPm ## Why Statistics in Psychology? * Psychology is a Science * Statistics can help us prove or disprove theories / constructs in psychology * Statistics can help us...

Here is the transcription of the images into markdown format: ## A STATISTICS REFRESHER Prepared by: Jan Mari B. Mangalili, RPm ## Why Statistics in Psychology? * Psychology is a Science * Statistics can help us prove or disprove theories / constructs in psychology * Statistics can help us construct measures or tests that will measure psychological constructs such as intelligence, personality aptitude, traits, and etc. ## Scales of Measurement | | | | :-------------- | :--------------------------------------------------------------------------------------------------------------------------------- | | **MEASUREMENT** | **SCALES** | | | | | the act of assigning numbers or symbols to characteristics of things (people, events, whatever) according to rules. | a set of numbers (or other symbols) whose properties model empirical properties of the objects to which the numbers are assigned. | | | | | :-------------------- | :---------------------------------------------------- | | **DISCRETE SCALE** | **CONTINUOUS SCALE** | | | | | scale used to measure a variable which cannot be theoretically divided. | scale used to measure a variable which can theoretically be divided. | *Error refers to the collective influence of all of the factors on a test score or measurement beyond those specifically measured by the test or measurement.* * In simple words, this is the degree to which the test scores/measurement may be wrong, considering other factors like state of the test takers (sickly or healthy), the venue of the test, the test itself and etc. ## Different Scales of Measurement | | | | :-------------------- | :----------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | **Nominal Scales** | **Ordinal Scales** | | | | | These scales involve **classification or categorization** based on one or more distinguishing the characteristics, where all things measured must be placed into mutually exclusive and exhaustive categories. | Like nominal scale, it permits categorization. The difference is, there is a **rank ordering** for each characteristics / response. | | | | | :-------------------- | :------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | | **Interval Scales** | **Ratio Scales** | | | | | It has the features of nominal and ordinal scales plus, it contains equal intervals between variables. It has **no absolute zero point**; it is theoretically possible to have negative values in this scale. | It has all the properties of nominal, ordinal, and interval and **has a true zero point**. Meaning, id the scale points to zero, the variable does not exist. | ## Measurement Scales in Psychology * Majority of Psychological tests are ordinal in nature; there is always a rank order in those measures * However, most tests often present themselves as interval variables because it is easier to describe interval variables using measures of central tendency, variability and their norms. ## Descriptive Statistics This is the field in statistics that is used to describe our data. * Raw Scores - the unmodified accounting of performance of the test-taker in a test * Frequency Distribution - a table of scores and how many times they occurred. They can be presented in several ways. Table 3-1 Data from Your Measurement Course Test | Student | Score (number correct) | | :---------- | :--------------------- | | Judy | 78 | | Joe | 67 | | Lee-Wu | 69 | | Miriam | 63 | | Valerie | 85 | | Diane | 72 | | Henry | 67 | | Esperanza | 92 | | Paula | 94 | | Martha | 62 | | Bill | 61 | | Homer | 44 | | Robert | 66 | | Michael | 76 | | Jorge | 87 | | Mary | 83 | | "Mousey" | 42 | | Barbara | 82 | | John | 84 | | Donna | 51 | | Uriah | 69 | | Leroy | 61 | | Ronald | 96 | | Vinnie | 73 | | Bianca | 79 | ## Frequency Distribution Table 3-2 Frequency Distribution of Scores from Your Test | Score | f (frequency) | | :---- | :------------ | | 96 | 1 | | 94 | 1 | | 92 | 1 | | 87 | 1 | | 85 | 1 | | 84 | 1 | | 83 | 1 | | 82 | 1 | | 79 | 1 | | 78 | 1 | | 76 | 1 | | 73 | 1 | | 72 | 1 | | 69 | 2 | | 67 | 2 | | 66 | 1 | | 63 | 1 | | 62 | 1 | | 61 | 2 | | 51 | 1 | | 44 | 1 | | 42 | 1 | ## Group Frequency Distribution Table 3-3 A Grouped Frequency Distribution | Class Interval | f (frequency) | | :------------- | :------------ | | 95-99 | 1 | | 90-94 | 2 | | 85-89 | 2 | | 80-84 | 3 | | 75-79 | 3 | | 70-74 | 2 | | 65-69 | 5 | | 60-64 | 4 | | 55-59 | 0 | | 50-54 | 1 | | 45-49 | 0 | | 40-44 | 2 | The first image is a histogram representing score distribution grouped by intervals of 5, from 41-45 to 96-100. The y-axis is labeled as "Number of cases". The second image shows a score distribution separated by Pass or Fail. There are 17 cases of pass and 5 of fail. ## Measure of Central Tendency | | | | :------------------------ | :--------------------------------------------------------------------------------------------------------------------------- | | | | | **ARITHMETIC MEAN** | which is referred to in everyday language as the “average”. The mean considers the actual numerical value of every score. | | **MEDIAN** | The median is defined as the middle score in a distribution, and is another commonly used measure of central tendency. | | **MODE** | The most frequently occurring score in a distribution of scores. | | | | | A statistic that indicates the average or midmost score between the extreme scores in a distribution. | | ## Variability * An indication of how scores in a distribution are scattered or dispersed. ## Measure of Variability | | | | :-------------- | :-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | | | | **RANGE** | The range of a distribution is equal to the difference between the highest and the lowest scores. | | **INTERQUARTILE** | Difference between the third and first quartiles of distribution. | | **SEMI-INTERQUARTILE** | The interquartile range divided by 2. | | | | | Statistics that describe the amount of variation in a distribution. | | | | | | :-------------- | :--------------------------------------------------------------------------------------------------------------------------------- | | | | | **STANDARD DEVIATION** | a measure of variability equal to the square root of the average squared deviations about the mean. More succinctly, it is equal to the square root of the variance | | **VARIANCE** | it is equal to the arithmetic mean of the squares of the differences between the scores in a distribution and their mean. | | | | | Statistics that describe the amount of variation in a distribution. | | ## SKEWNESS | | | | :-------------------------------------------------------------------------------------------------------------------------------- | :----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | | | | **POSITIVE** | it is when relatively few of the scores fall at the high end of the distribution. Positively skewed examination results may indicate that the test was too difficult. More items that were easier would have been desirable in order to better discriminate at the lower end of the distribution of test scores | | **NEGATIVE** | it is when relatively few of the scores fall at the low end of the distribution. Negatively skewed examination results may indicate that the test was too easy. In this case, more items of a higher level of difficulty would make it possible to better discriminate between scores at the upper end of the distribution. | | Distributions can we characterized by their skewness, or the nature and extent to which symmetry is absent. It is an indication of how the measurements in a distribution are distributed. | | The image shows three curves. The first one is labeled Positive Skew and Median and Mode are to the right of Mean. The second one is labeled Symmetrical Distribution, and the Mean, Median, and Mode are at the same location. The third one is labeled Negative Skew, and the Mean, Median, and Mode are to the left of Mean. ## KURTISOS | | | | :-------------- | :--------------------------------------------------------------------------------- | | | | | **PLATYKURTIC** | relatively flat; scores are distributed almost equally through the higher and lower end. | | **LEPTOKURTIC** | Relatively peaked; scores are mostly in the center, with fewerscores in the higher and lower end. | | **MESOKURTIC** | somewhere in the middle; the ideal kurtosis | | | | | refer to the steepness of a distribution in its center | | The image is of a graph with three curves plotted on it. The curves start at -3 and end at +3 in the X axis labeled z scores. The darkest curve is labeled Leptokurtic. The flattest curve is labeled Platykurtic. The curve in between is labeled Mesokurtic. ## The Normal Curve * a bell-shaped, smooth, mathematically defined curve that is highest at its center. From the center, it tapers on both sides approaching the X-axis asymptotically (meaning that it approaches, reaches, or touches, the axis). * In theory, the distribution of the normal curve ranges from negative infinity to positive infinity. * The curve is perfectly symmetrical, with no skewness. * Because it is symmetrical, the mean, the median, and the mode all have the same exact value. The image shows graph of the normal curve indicating how the data is distribuited: * ±3σ ≈99.7% * ±2σ ≈95% * ±1σ ≈68% The x axis represents Raw scores, and the z zones go from 5(-3σ) to 95(3σ) ## STANDARD SCORES | | | | :-------------- | :--------------------------------------------------------------------------------------------------------------------------- | | | | | **z Scores** | results from the conversion of a raw score into a number indicating how many standard deviation units the raw score is below or above the mean of the distribution | | **T-scores** | composed of a scale that ranges from 5 standard deviations below the mean to 5 standard deviations above the mean | | **Stanine** | a term that was a contraction of the words standard and nine. | | | | | a raw score that has been converted from one scale to another scale, where the latter scale has some arbitrarily set mean and standard deviation. | | The image shows graph of the normal curve indicating how the data is distribuited: *z-Scores (μ = 0, σ = 1) : goes from -4 to +4 *IQ Scores (μ = 100, σ = 15): goes from 40 to 160 *T Scores (μ = 50, σ = 10): goes from 10 to 90 *Scaled Score (μ = 10, σ = 3): goes from 1 to 19 *Percentile Rank : goes from 0.003 to 99.997 ## CORRELATION & INFERENCE | | | | :---------------------------------- | :--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | | | | **CORRELATION** | One of the most basic way to make an inference in a given data; an expression or degree of correspondence between two things. | | **CORRELATION COEFFICIENT** | A numbers provides us with an index of the strength of the relationship between two things. It ranges from -1 to 1 | | | | ## REFERENCE * Cohen, R. J., Schneider W. J., & Tobin R. M (2022). Psychological Testing and Assessment: An Introduction to Tests and Measurement (10th ed.). McGraw-Hill Education.