Calculating & Reporting Healthcare Statistics Second Edition Chapter 10 Descriptive Statistics in Healthcare

Descriptive Statistics Used to explain data in ways that are manageable and easily understood ©2006 All rights reserved.

Descriptive Statistics Rank Denotes a score’s position in a group relative to other scores organized in order of magnitude The position of the observation is more important than the number associated with it ©2006 All rights reserved.

Descriptive Statistics Quartile Data organized in order of magnitude divided into four equal parts – each part is a quartile First quartile is the lowest 25% of the data Second quartile is the is up to 50% of the data Third quartile is up to 75% of the data Fourth quartile includes the remaining 25% of the data ©2006 All rights reserved.

Descriptive Statistics Decile Represents data divided into ten equal parts First decile is the lowest 10% of the data The ninth decile includes the first 90% of the scores ©2006 All rights reserved.

Descriptive Statistics Percentile Separate the distribution into 100 equal parts A person who scores at the 54th percentile has a score greater than or equal to 54% of all the scores in the distribution Called a percentile rank ©2006 All rights reserved.

Descriptive Statistics Percentiles Help people understand their score relative to all scores from a group ©2006 All rights reserved.

Descriptive Statistics Measures of Central Tendency In summarizing data, it is often useful to have a single typical or average number that is representative of the entire collection of data or specific population Three measures of central tendency are frequently used: mean, median, and mode ©2006 All rights reserved.

Descriptive Statistics Frequency Distribution Shows the values that a variable can take and the number of observations associated with each value A variable is a characteristic or property that may take on different values ©2006 All rights reserved.

Descriptive Statistics Mean The arithmetic average It is common to use the term “average” to designate mean To obtain the mean, add all the values in a frequency distribution and then divide the total by the number of values in the distribution For example, seven hospital inpatients have the following lengths of stay: 2, 3, 4, 3, 5, 1, and 3 The frequency distribution in order is 1, 2, 3, 3, 3, 4, and 5 To construct a frequency distribution, all the values that the LOS can take are listed and the number of times a discharged patient had that particular LOS is entered ©2006 All rights reserved.

Descriptive Statistics Mean In order to determine the mean we sum all the values in the frequency distribution and divide by the frequency The total is 21 We arrived at this by adding 2 + 3 + 4 + 3 + 5 + 1 + 3 To arrive at the mean, divide 21 by the number of values (or frequency distribution) which in our case is 7 The mean (or average) equals 3 days Formula Total sum of all the values / Number of the values involved = X OR Σ scores/ N ©2006 All rights reserved.

Descriptive Statistics Median The midpoint (center) of the distribution of values It is the point above and below which 50 percent of the values lie Describes the middle of the data, literally The median value is obtained by arranging the numerical observations in ascending or descending order and then determining the value in the middle of the array This may be the middle observation (if there is an odd number of values) or a point halfway between the two middle values (if there is an even number of values). To arrive at the median in an even-numbered distribution, add the two middle values together and divide by 2 The advantage of using the median as a measure of central tendency is that it is unaffected by extreme values ©2006 All rights reserved.

Descriptive Statistics Median May be used in reporting data instead of the mean Provides a more revealing representation of the data when there are outliers in the distribution ©2006 All rights reserved.

Descriptive Statistics Mode The value that occurs with highest frequency The most typical It is the simplest of the measures of central tendency because it does not require any calculations In the case of a small number of values, each value likely may occur only once and there will be no mode The mode is rarely used as a sole descriptive measure of central tendency because it may not be unique because there may be two or more modes These are called bimodal or multimodal distributions ©2006 All rights reserved.

Descriptive Statistics The choice of a measure of central tendency depends on the number of values and the nature of their distribution Sometimes the mean, median, and mode are identical The mean is preferable because it includes information from all observations If the series of values contains a few that are unusually high or low, the median may represent the series better than the mean The mode is often used in samples where the most typical value is preferred The mode does not have to be numerical ©2006 All rights reserved.

Descriptive Statistics Measures of Variation Variability We also want to consider the spread of the distribution, which tells us how widely the observations are spread out around the measure of central tendency The mean gives a measure of central tendency of a list of numbers but tells nothing about the spread of the numbers in the list Variability refers to the difference between each score and every other score ©2006 All rights reserved.

Descriptive Statistics Range The simplest measure of spread The range is the difference between the largest and smallest values in a frequency distribution The easiest to compute It is the simplest, order-based measure of spread, but it is far from optimal as a measure of variability for two reasons First, as the sample size increases, the range also tends to increase Second, it is obviously affected by extreme values which are very different from other values in the data ©2006 All rights reserved.

Descriptive Statistics Variance A frequency distribution is the average of the standard deviations from the mean The variance of a sample is symbolized by s2 The variance of a distribution is larger when the observations are widely spread ©2006 All rights reserved.

Descriptive Statistics Variance To calculate the variance, first determine the mean Then, the squared deviations of the mean are calculated by subtracting the mean of the frequency distribution from each value in the distribution. The difference between the two values is squared (X – Xbar )2 The squared differences are summed and divided by N – 1 S2 = variance  = sum X = value of a measure or observation Xbar = mean N = number of values or observations The term N – 1 is used in the denominator instead of N to adjust for the fact that the mean of the sample is used as an estimate of the mean of the underlying population ©2006 All rights reserved.

Descriptive Statistics Standard Deviation The standard deviation is kind of the "mean of the mean“ Standard deviation (SD) is the square root of the variance Because SD is the square root of the variance, it can be more easily interpreted as a measure of variation If the SD is small, there is less dispersion around the mean If the SD is large, there is greater dispersion around the mean ©2006 All rights reserved.

Descriptive Statistics Standard Deviation To understand this concept, it can help to learn about what mathematicians call normal distribution of data A normal distribution of data means that most of the values in a set of data are close to the "average," while relatively few values tend to one extreme or the other The standard deviation is a statistic that tells you how closely all the observations are clustered around the mean in a set of data When the examples are pretty closely gathered and the bell-shaped curve is steep, the standard deviation is small When the examples are spread apart and the bell curve is relatively flat, that tells you have a relatively large standard deviation ©2006 All rights reserved.

Descriptive Statistics Standard Deviation Normal distribution means that if the variable on every person in the population were measured, the frequency distribution would display a normal pattern, with most of the measurements near the center of the frequency It also would be possible to accurately and summarily describe the population, with respect to variable, by calculating the mean, variance, and SD of the values In a normal distribution, one SD in both directions from the mean contains 68.3 percent of all values Two SDs in both directions from the mean contain 95.5 percent of all values Three SDs in both directions from the mean contain 99.7 percent of all observations ©2006 All rights reserved.

Descriptive Statistics Not all distributions are symmetrical or have the usual bell-shaped curve Some curves are skewed Their numbers do not fall in the middle, but rather on one end of the curve Skewness is the horizontal stretching of a frequency distribution to one side or the other so that one tail is longer than the other The direction of skewness is on the side of the long tail That is, if the longer tail is on the right then the curve is skewed to the right If the longer tail is on the left, then the curve is skewed to the left ©2006 All rights reserved.

Descriptive Statistics Correlation Measures the extent of a linear relationship between two variables Can be described as strong, moderate, weak, positive (direct) or negative (inverse) A correlation of 0 means there is no relationship between the variables ©2006 All rights reserved.