Nurs 211 - Stats Lecture 2 Notes

Lecture 2 - Sept 18 Levels of measurement Level of Measurement Central Tendency (the way we measure the central value) Graphs Nominal Mode Bar, pie Ordinal Mode, median Bar, pie Interval Mean, mode, median Bar, pie, box plot, histogram Ratio Mean, mode, median Histogram, box plot Key concepts - Sampling bias: Sampling bias occurs when some members of a population are systematically more likely to be selected in a sample than others. (systematic pressure/issue) - Error: always present in a sample (unavoidable) (underlying) (humans make mistakes) Effect size: it is a statistical concept that measures the strength of a relationship between two variables on a numeric scale (how big a difference between 2 groups) - Confounding variable: third variable in a study examining a potential cause and effect relationship. A confounding variable is related to both the supposed cause and the supposed effect of the study. - Ex. Smoking will mess with the outcome of treatments. Eliminate it to guarantee more accurate results Stratified sampling vs cluster sampling Descriptive statistics - Descriptive statistics are concerned with the presentation, organization and summarization of data - Graphical representation and tables - Typically the first thing a health researcher does - Includes demographic data (age, smoking status, comorbidities, education, ect) Often describes in a table at the beginning of research papers - Make sure the healing speed is due to the type of dressing and not the size of the wound Descriptive statistics (summary statistics): measures of central tendency Given a data set, a measure of the central tendency is a value about which the observations tend to cluster - OR - It is a value around which a data set is centered - Is there a number the data tends to cluster around? Central tendency - Three main measures - Mean (average) - Mode - Median Examples of how central tendency is used in nursing - Average medication dose based on patient's height and weight (needs to be calculated, especially with children and infants) - Average patient satisfaction scores - Infection rates to monitor the effectiveness of infection control practices and interventions - Median age of patients in a nursing home - Mean blood pressure readings of a group of patients to assess the overall cardiovascular health of the group Mean - The average score (take a meaningful average) - Used for variables measured at the interval and ratio level - Calculate by adding up the numbers in a distribution and then divide by the number of observations (sum of all items divided by the number of items) Median - The middle value in an ordered array - Used for variables measured at the ordinal, interval or ratio level - Steps to find the median 1. List the values from lowest to highest (put them in order) 2. Find the middle score (n+½) 3. For eve number of numbers, take the average of the two middle scores Mode - The most frequent category - Used for variables measured at the nominal (often used) level, ordinal interval and ratio Distributions can have more than one mode Mode, mean and median shown graphically (positive, negative and symmetrical) Bimodal distribution (two of more central clusters) No mode Measures of variability - Measures of variability describe the spread or the dispersion of a set of data We can see how much our data varies from the mean - Common measures of variability - Range - Variance - Standard deviation Some examples for nurses - Variations in physical assessment - help identify risk assessment and help nurses and healthcare teams assess the risk of complications and make timely interventions - Vital signs and lab results variability - help nurses understand the range of values for different patients, allowing them to tailor interventions and care plans accordingly - Variability in patients admissions - understanding the variability in patient admissions helps hospitals allocate staff and beds effectively - Variations in care processes and outcomes - helps in quality improvement initiatives Range - How far apart scores are from the mean - The range = the highest value - the smallest value - Usually the smallest and largest values are reported - Ex. if the highest value is 48 and the smallest value is 35, the range could be reported as 13 OR 35-48 - The - is not a minus sign, it means to - Basically to get the range, minus the biggest number from the smallest Variance and standard deviation - Variance - average distance of the values from the arithmetic mean (how spread out is the data) - OR - The measure of dispersion or spread around the mean. Gives info on the average degree to which each data point differs from the mean - Standard deviation - is defined as the square root of the variance and it gives info on how far the responses/numbers are spread from the mean - Standard way to look at variance around the mean Variance (around mean) Formula for calculating the variance and standard deviation (don’t actually need this, just for people who want to know) A degree of freedom is a way of adjusting for sample size Introduction to inferential statistics - Need interval or ratio level data - Key concepts that underpin inferential statistics - Probability - Normal distribution - Population and sample - Standard scores (z-scores) Probability - Need to assign probabilities to each variable - How likely an event is to occur - Quantified as a number between 0 and 1 (close to 0 as we can get) 0 = zero chance - 0.5 = even chance - 1 = certain to occur - Higher the probability the more certain the event is to occur - Probability = the number of possibilities divided by the total number of outcomes Ex. chance of rolling a 6 with one dice - Number of ways of rolling a 6 = 1 - Total number of possible outcomes = 6 - So the probability is ⅙ or 0.17 Sample size and population - As the number of people in the sample goes up: - The mean of the sample gets closer to the mean of the population - The distribution of a variable gets closer to a normal distribution - Can occur somewhere around 30 observations/people - When we think about drawing conclusions from a sample to the whole population, we are relying on the assumption that the data are distributed normally - Most people are average… What is the normal distribution - A visual representation of a group of scores - It is a probability distribution - Probability is above or below the average - Y-axis = number of people/frequency - X-axis = variable of interest (ex. Weight, blood glucose, blood pressure) Characteristics of normal distribution - Bell shaped curve - Most of the data close to the mean ( most is in the largest part) - Symmetrical with mean in the middle - The tails come closer and closer to the horizontal axis but never touch (never hit 0, just get closer to infinity) - Total area under the curve is 1 (100%) - Highest probability of a data point is in the middle (most people are average) - The numbers on the x-axis refer to standard deviations from the mean - The area under any portion of the curve is the probability of x being within that span - The area under the curve between -1SD and +1SD is 0.682, thus the probability that an x value is between 1 standard deviations +- is 68.2% The 68-95-99.7% rule - Shorthand to remember the % of values around the mean (either side of the mean for normally distributed data - ± 1 s = about 68% - ± 2 s = about 95% - ± 3 s = about 99% - Plus or minus _ standard deviations Central limit theorem - If a sample is large enough, the distribution of probable outcomes takes the shape of the normal distribution, sometimes called the bell curve = central limit theorem - The bigger the sample, the more likely the data will have a normal distribution - Central limit theorem is vital in nursing for ensuring the validity of statistical analyses, making accurate inferences from sample data, and supporting data-driven decision making in various aspects of patient care, research and quality improvement - The mean and median are equal - Data is symmetrical around the center - 50% of the values greater than the mean - 50% of the values less then the mean - It is the most important and widely used distribution in statistics - Parametric tests are better and more powerful. They’re based on mean and normal distribution - Researchers cheat their data so they can use them - Nonparametric tests are not so good and are based on median and rank For every parametric test, there is a nonparametric equivalent - The more people there are in a sample, the more the mean of the sample will look like the wider population - The more chance that the variables of interest will be normally distributed The variance decreases - The less variance means a more normal distribution - As the sample size increases, just about any distribution will tend to become normally distributed - Use a parametric test if you know the sample is normally distributed - If it’s not of you don’t know, use a non parametric test Shapes of distribution - Three common shapes of distributions are: - Normal: bell shaped curve - symmetrical - Skewed: non-normal - non symmetrical - can be positively or negatively skewed (values tail off above/below the mean) - Multimodal: has more than one peak (mode) (large vs small dragon wings) Kurtosis - Distributions can also differ from a normal distribution in terms of how steep the curve (its pointiness) - How pointy or flat the curve is - Leptokurtic - many scores in the tails (narrow) - Platykurtic - less scores in tails and flatter Standard normal distribution - One of the reasons why normal distributions are valuable is that they make it possible to standardize scores - These standardized scores are called z-scores - Z-scores allow the researcher to determine the probability of the occurrence of a particular score - Z-scores also make it possible to compare scores from different measures - Z-score (a way of quantifying differences) of a measurement X indicates how many standard deviations is the measurement away from the mean? A positive z-score indicates that the measurement is above the mean and a negative z-score indicated that the measurement is below the mean - The z-score is a variable that has a normal distribution with mean X=0 and standard deviation s=1. This distribution is called standard normal distribution Z-scores - Also known as the standard score - Use the normal curve to convert scores on different tests into standard scores so that they can be compared - Z-score is a measure of how many SD above or below the mean - Range from -4 SD to +4 SD - Also tell you where a particular score is in relation to the mean - Matches group mean Z score = 0 - Above group mean = +ve - Below group mean = -ve Value of z-scores - Calculate by 1. Value (X) - mean 2. Divide by SD - Allows comparison of observed distribution to expected distribution ex. If normally distributed Applying z-scores to health - Lab tests - Criteria for deciding if results of glucose, thyroid, electrolytes, etc, are abnormal based on the normal distribution - Bone density - Results based on density for age, sex, weight, and ethnic or racial origin - Z-score of -2 or lower could suggest that something other than aging is causing abnormal bone loss - Growth charts for children and babies - Scores that are abnormal are defined by those that occur less than 95% of the time (< -2SD and > +2SD) - Check the blood glucose level of hundreds of people, take the average and get ±2 standard deviations - Depression example - If the scores are very different, you’re not measuring depression well One test might be goo and the other is bad or they both might suck Outliers - In a normal distribution any measurement with a Z score of greater then 3 or less than -3 is called an outlier - Note: an outlier is an extreme or unusual measurement, recall that the Empirical Rule states that for a normal distribution 99.9% of the measurements fall within 3 standard deviations around the mean