Descriptive Statistics, Central Tendency, Variation PDF

STA 683 Descriptive Statistics, Normal Curve, Percentiles, Probability, Central Tendency & Variation Dr. Jessica Westman, PhD, RN  1. EXAMINE DESCRIPTIVE STATISTICS. 2. DISPLAY DESCRIPTIVE STATISTICS. 3. DESCRIPTIVE A FREQUEN...

STA 683 Descriptive Statistics, Normal Curve, Percentiles, Probability, Central Tendency & Variation Dr. Jessica Westman, PhD, RN  1. EXAMINE DESCRIPTIVE STATISTICS. 2. DISPLAY DESCRIPTIVE STATISTICS. 3. DESCRIPTIVE A FREQUENCY DISTRIBUTION. 4. CALCULATE CUMULATIVE FREQUENCY AND PERCENTAGES FOR A GROUP OF DATA. 5. COMPARE AND CONTRAST DESCRIPTIVE AND INFERENTIAL STATISTICS. Learning 6. DEFINE AND DISTINGUISH BETWEEN THE MEAN, MEDIAN, MODE, AND STANDARD DEVIATION. 7. IDENTIFY UNIMODAL, BIMODAL, AND MULTIMODAL. 8. DEFINE PROBABILITY AND EXPLAIN THE RANGE OF POSSIBLE PROBABILITIES. Objectives 9. 10. 11. COMPARE AND CONTRAST FREQUENCY AND PROBABILITY DISTRIBUTIONS. CONTRAST POSITIVE AND NEGATIVE DISTRIBUTION SKEWS AND DESCRIBE WHERE OUTLIERS ARE PRESENT. EXAMINE AND DESCRIBE NORMAL DISTRIBUTIONS. 12. UNDERSTAND DISTRIBUTION, CENTRAL TENDENCY, AND VARIABILITY WITHIN NORMAL AND SKEWED DISTRIBUTIONS. 13. EXAMINE PROBABILITY FROM MULTIPLE APPROACHES. Frequencies & Percentages Frequency: the number of times a score or value for a variable occurs in a set of data Used to describe demographic and study variables measured at nominal or ordinal levels Percentage: a portion or part of the whole or a named amount in every hundred measures 3 Frequency Distribution This describes the distribution of the variables Basic way to organize data. Summarizes the occurrences of events under study; tallies the frequency of events. Cohort groups are sometimes created to investigate the frequencies of certain data. Example is for discrete (nominal or ordinal) data 4 Percentiles Often used in healthcare The percentage of cases a given score exceeds Median is the 50th percentile A score in the 90th percentile is exceeded by only 10% of scores 5 Probability Probability of one event happening (below), of two (above) Probability of two or more Helps you make decisions independent events Simple examples: rolling a die, flipping a coin More complex: weather, the probability of a person developing a disease What is the probability of rolling an even number on one regular die? What is the probability of rolling a 5 on one regular die? If you roll a 5 this time, does that change the probability of you rolling a 5 next time? What is the probability of choosing a 4 in a deck of cards? If you choose a 4 the first time, does this change the probability of the second choice? Probability of a single event occurring 6 Central Tendency Summarizes the middle of the group. Each measure has specific uses and is most appropriate to select types of distribution and measurement. Mode: most frequent score An “average” Median: middle score Mean: average score 7 Measures of Central Tendency: Mean Most common measure of central tendency “μ” for the mean of a population, M for the mean of a sample The sum of a scores divided by the number of scores Can be used for interval, and ratio level data Sensitive to outliers 8 Outliers A value in a sample Can result in skewed data set that is distributions and can unusually low or interfere with high statistical analysis 9 Measures of Central Tendency: Median The midpoint of a distribution, scores have to be in order Also known as the 50th percentile When there is an odd number of numbers, the median is the middle number When there is an even number of numbers, the median is the mean of the two middle numbers (add together and divide by two) Can be used with ordinal (once coded), interval, and ratio data 10 Measures of Central Tendency: Mode Most frequently occurring value Can be used for nominal, ordinal, interval, and ratio level data, but most often used with nominal Bimodal: when a distribution has 2 modes Multimodal: when a distribution has 3 or more modes 11 The Normal/Bell Curve o Symmetric around their mean o The mean, median, and mode are the equal o Denser in the center and less dense in the tails o 1 SD: 68% of scores o 2 SD: 95% of scores o 3 SD: 99.7% 12 Measures of Variability How spread out the data is 4 frequently used measures of variability: 1. Range 2. Interquartile range (covered earlier with percentiles) 3. Variance (the average squared difference of the scores of the mean, look at page 148 in Lane., used more frequently when calculating SD by hand) 4. Standard deviation 13 Interpret Descriptive Stats 1) 68% of infants’ gestational ages (group 1) were between ___ & ___ 2) 95% of infants’ birthweights (group 1) were between ___ & ___ 3) 99.7% of infants’ postnatal ages (group 1) were between ___ & ___ Measures of Variability: Range Simplest measure of variability The highest score minus the lowest score Can also be reported in research as: The age of participants ranged from 18-62 15 Measures of Variability: Standard Deviation Most frequently used measure of variability The square root of the variance, the average number of points by which the scores of a distribution vary from the mean Average deviation of scores from mean Always reported with the mean When the SD is larger, this means there is more variability in the scores Example: Score for the test was 75% with a standard deviation of 3.1%. Let’s work out the 1st, 2nd, and 3rd SDs and what would be outside of it. 16 Standard Deviation by hand 1. Calculate the mean of the values 2. Take each value minus the mean 3. Square each value 4. Add squared values 5. Divide by n-1 6. This is now the variance 7. Square root of variance to get standard deviation 17  Let’s Practice Values: 3, 5, 2, 8, 10, 14 18  Z-Score: Standardizing Distributions A standardized distribution is composed of scores that have been transformed to create predetermined values for the mean and SD. These standard distributions are used to make dissimilar distributions comparable z scores tell us exactly where an individual score is located Therefore, makes it possible to compare different scores or different individuals even though they may come from completely different distributions To do this, individual scores are converted to standard scores (reported in SD units) z = (score-mean)/SD Normal Distributions with different standard deviations 20 Reporting in Research 21 What if the data is not normally distributed? Best way to see this is using a graph (bar graph, line graph) Statistical software will also let you know this information Should be assessed prior to statistical analysis (Shaprio-Wilk’s W test) 22 Skewness Positive skew: a longer tail in the positive direction, also described as skewed to the right. Example: World income Negative skew: a longer tail in the negative direction, also described as skewed to the left. Example: Age of death 23 Kurtosis (Peakness) Leptokurtic- extremely peaked (above 0) Mesokurtic- normal (0) Platykurtic- Flat (below 0) 24 Review Question Which measure of central tendency always represents an actual score of the distribution? Provide a rationale for your answer. a. Mean b. Median c. Mode d. Range 25 Review Question 10 ACS patients were asked to rate their pain in their jaw and neck on a 0–10 scale: 3, 4, 7, 7, 8, 5, 6, 8, 7, 9. What are the range, median, mode, and mean for the pain scores? 26 Review Question The number of nursing students enrolled in a particular nursing program between the years of 2013 and 2020, respectively, were 563, 593, 606, 540, 563, 610, and 577. Determine the mean (M), median (MD), and mode of the number of the nursing students enrolled in this program. 27 Is this data negatively or positively skewed? 28 Variability (more or less) 29