PSYCH 105 Chapter 2a&b PDF

PSYCH 105 Chapter 2: DESCRIPTIVE STATISTICS Course Presentation Prepared by: Soujee Ann W. Mangapac Dep’t of Psychology Faculty BSS – CSS  DESCRIPTIVE Statistics Descriptive statistics are used to summarize and describe the main features of a dataset. Types of Descriptive Statistics: Mea...

PSYCH 105 Chapter 2: DESCRIPTIVE STATISTICS Course Presentation Prepared by: Soujee Ann W. Mangapac Dep’t of Psychology Faculty BSS – CSS  DESCRIPTIVE Statistics Descriptive statistics are used to summarize and describe the main features of a dataset. Types of Descriptive Statistics: Measures of Central Tendency (Mean, Median, Mode) Measures of Dispersion (Range, Variance, Standard Deviation) Measures of Shape (Skewness, Kurtosis) Statistics Fun Facts Did you know? Excerpts from: https://www.did-you-knows.com/did-you-know-facts/statistics.php?page=26.......... 11% of people are left-handed..........August has the highest percentage of births in a year........... Koalas sleep an average of 18 hours a day........... An average person will spend 25 years asleep........... The most commonly forgotten item for travelers is their toothbrush........... Your most sensitive finger is your index finger (closest to your thumb)...........The average golf ball has 336 dimples........... Monopoly is the most-played board game in the world........... A piece of paper cannot be folded more than 7 times........... Hiccups usually last for 5 minutes........... Your most active muscles are in your eye. Measures of Central Tendency A measure of the central tendency of a set of data gives us an indication of the typical score in that dataset. In more technical terms, it’s about the point where most of the data converges. In this sense, its position is not absolute. If visualized, it may be present at the center, the left, the right, or practically anywhere. PSYCH 105: Chapter 2b Measures of Central Tendency & Variability for Ungrouped Data Measures of Central Tendency for Ungrouped Data 1. Arithmetic Mean (μ or x) Arithmetic mean, or simply mean, is represented by the μ (“mew”) symbol when referring to population mean and the x¯ (“x-bar”) in samples. It is the sum of all scores divided by the total number of scores It is the most sensitive measure of central tendency and responds to every change in the entire distribution. 1. Arithmetic Mean (μ or x) Measures of Central Tendency Properties of the Mean This is the value from which the scores deviate in such a way that their algebraic sum always equals to zero. Measures of Central Tendency for Ungrouped Data 2. Median the value that lies in the middle of the sample In a data set with an odd total count (e.g., n = 7), the 4th value is the Median. In a data set with an even total count (e.g., 8), the Median is computed by taking the average of the two values in the middle. The median is a positional measure, which means that it is not influenced by extreme values in the data set. For example, given this data set: Distribution A: 4, 6, 10, 2, 13, 12, 11 Distribution B: 11, 4, 6, 10, 32, 45, 8 The numbers are arranged in ascending order. 1. Thus, it will be arranged as follows: 2. 3. Distribution A: 2, 4, 6, 10, 11, 12, 13 4. Distribution B: 4, 6, 8, 10, 11, 32, 45 5. 6. As you can see here, the median for both sets of data is 10. Notice that since the median is a positional measure, the presence of extreme scores in Distribution 2 (32 and 45) does not affect the value of the median. 7. 8. Even Set of Data Example: 9. 2, 4, 6, 9, 10, 11, 12, 13 10. Thus, 9 + 10/2 = 9.5 11. Keep in mind that this is NOT AN ACTUAL VALUE, but rather an ESTIMATED VALUE. Measures of Central Tendency for Ungrouped Data 3. Mode ✓ most frequently occurring score or is the most “typical score”; simplest measure of central tendency The mode is usually unstable, especially in small groups. It only has a significant meaning with a large mass of data. Because of this, the mode is the least used measure of central tendency. Measures of Central Tendency Identify the most frequently occurring value. Data Set 1: 34 36 42 45 47 50 52 54 35 36 38 42 42 49 53 55 Data Set 2: 34 35 36 38 42 43 44 47 49 52 53 55 57 58 59 61 Measure Advantage Disadvantages Scale of Measureme nt Mean Single & unique value Affected by a few Interval Representative (every score is extreme scores Ratio entered in its computation) Most stable May be used in further computations Median More stable from group to Not necessarily Ordinal group than mode representative Interval Cannot be used in further Ratio analyses because it is not computed from the scores themselves but from their order Mode Easy to obtain Not necessarily Nominal representative Ordinal May not yield a unique Interval Measures of VARIABILITY or DISPERSION 1. Range The difference between the maximum and minimum values in the dataset. Use: Provides a simple measure of spread, but it is sensitive to outliers. Range = Max Value−Min Value 2. Variance (σ^2 or s^2) Variance is the average of the squared differences from the mean. It gives an idea of the spread of the data points. Variance, to some extent, serves as a better approximation of the average distance between each individual score from the mean. The divisor for the population SD and population variance is only N or n instead of n-1 or N-1. The expression “N-1” refers to Degrees of Freedom. If the sum of squares is equal to 30 and the n is 20, what is the population parameter in terms of variance? Answer: 3 3. Standard Deviation (σ or s / sd) the standard deviation is represented by the Greek letter, sigma, in populations and a small letter s or sd in samples the SD provides an estimate of the average distance of each data point from the mean of the dataset 3. Standard Deviation (σ or s / sd) Formula for Population SD Formula for Sample SD 3. Standard Deviation (σ or s / sd) A higher standard deviation indicates that the data points are more spread out from the mean, while a lower standard deviation indicates that they are closer to the mean. Standard deviation gives more weight to outliers due to squaring the differences from the mean. Conclusion This tells us that data values are distant from the mean at an estimated value of 2.449. Assuming all other factors are constant, considering that the range is from 1 to 8, the mean is 4.5, and the total count is 8, the data is likely well spread. If the sum of squares is equal to 30 and the n is 10, what is the population parameter in terms of standard deviation? Ans: 1.73 If the sum of squares is equal to 30 and the n is 10, what is the sample statistic in terms of standard deviation? Ans: 1.83 If the sum of squares is equal to 30 and the n is 10, what is the sample statistic in terms of variance? Ans: 3.33 If the sum of squares is equal to 30 and the degrees of freedom is 19, what is the sample statistic in terms of variance? Ans: 1.58  Measures of DISPERSION/VARIABILITY/SPREAD for Sample Variance PSYCH 105: Chapter 2b Measures of Central Tendency & Variability for Grouped Data Course Presentation Prepared by: Soujee Ann W. Mangapac Dep’t of Psychology Faculty BSS – CSS  CI CB f x fx x-x̄ (x-x̄)^2