Chapter 6: Descriptive Statistics

Learning Objectives Describe statistical measures used in descriptive statistics Compute measures of central tendency Compute measures of variability Understand and choose the best measure of central tendency and variability for different levels of measurement Describe the normal distribution and associated statistics and probabilities Develop concepts of interval estimates and describe methods for determining sample size Apply understanding of central tendency and variability to nursing practice

Introduction In the previous chapter, we saw how nurses in practice can present data in a variety of formats such as graphs, charts and tables. However, we also lose some detail in the data when displayed graphically, especially around the distribution of data that are measured at the interval and ratio level (continuous variables). When the data are measured at the interval or ratio level, it is important to present the distribution of data in terms of central tendency and variability.

Descriptive Statistics Help us to explain the data more accurately and in greater detail. Help us to explain the distribution of data in terms of center and variability

Measures of Central Tendency What would be the typical score (average score) if I were to randomly select a person from a sample? Three common measures of central tendency – Mode – Median – Mean

Mode Mode is simply the most frequently occurring number in a given data set. The following data set represents 7 readings of systolic blood pressure: What is the mode of this sample?

Median Median is the exact middle value in a distribution, which divides the data set into two exact halves. The following data set represents income of 5 selected registered nurses: 35,000 39,50042,000 47,500 52,000 What is the median of this dataset?

Mean Mean is the sum of all data values in a data set divided by the number of data values and can be calculated by the following equation: The following data set contains 6 sodium content level measurements: What is the mean of this data set?

Choosing a measure of central tendency If you have nominal data, only use the mode Use the median when data are extremely skewed since it is not affected by extreme scores Otherwise, use the mean

Measures of Dispersion How well does a measure of central tendency represent the “middle/average” value in the data set? If I were to randomly select a person over and over again, would I get pretty close score to the measure of central tendency for the data set?

Measures of Variability Four common measure of variability – Range – Interquartile Range (IQR) – Variance – Standard Deviation

Range Range is simply the difference between the largest and the smallest values in the data set. The following data set contains measurements of pain level after vascular surgery from 8 patients on a scale of 1 to What is the range of this data set?

Range Range is simple to calculate, but one should be cautious about using range as a measure of variability as any outlier, either the smallest or the largest, can greatly distort it. Consider the following data set

Interquartile Range (IQR) IQR is the difference between 75 th percentile and 25 th percentile. It is less sensitive to an outlier(s) than range as it does not use the smallest and the largest value. What is the IQR for the following data set?

Variance Variance is average amount that data values differ from the mean and is computed with the following formula: What is the variance of the following data set of weight loss measurements of 5 patients?

Variance If you have calculated variance correctly, you should’ve gotten Note that the unit of this variance is the square of pound. What does (pound)² mean? This does not make sense!!! We need to bring the unit back to the original.

Standard Deviation Standard Deviation is the square root of variance. Taking square root of 5.84, we get 2.42 and this is how much data values differ from the mean on average.

Choosing a measure of variability Sensitivity to extreme values – Range – extremely sensitive – Standard deviation – very sensitive – Interquartile range – not sensitive Standard deviation – Has desirable statistical properties – Suggests numbers of cases in different intervals for bell-shaped distributions

Normal Distribution It is bell-shaped and symmetric. Area under a normal curve is equal to 1.00 or 100%. 68% of observations fall within 1 standard deviation away from the mean to both directions. 95% of observations fall within 2 standard deviations away from the mean to both directions. 99.7% of observations fall within 3 standard deviation away from the mean to both directions. Many normal distributions exist with different means and standard deviations.

Normal Distribution Normality can be checked visually with histogram. Statistical measures such as skewness and kurtosis can also be computed. – Skewness is a measure of whether the set is symmetrical or skewed. – Kurtosis is a measure of how peaked a distribution is. – If they fall within -1 to +1 range, the distribution is said to be “normal”. – Otherwise, the distribution is said to be non-normal.

Normal Distribution Why do we care about normal distribution? – Many human characteristics fall into approximately normal distribution and most of statistical analyses assume normal distribution of the variables.

Standard Normal Distribution Can we compare scores from two different distributions? – No, not if they are from different scales Therefore, we need to somehow put these two different distributions on a same scale for a legitimate comparison and Standard normal distribution is our solution.

Characteristics of Standard Normal Distribution The standard normal distribution has a mean of 0 and standard deviation of 1. Area under the standard normal curve is equal to 1 or 100% and z scores have associated probabilities, which are fixed and known. 68% of observations fall within 1 standard deviation away from the mean to both directions 95% of observations fall within 2 standard deviations away from the mean to both directions 99.7% of observations fall within 3 standard deviation away from the mean to both directions.

Characteristics of Standard Normal Distribution To convert a normal distribution to a standard normal distribution, we need to compute z- score using the following equation: When z-score is positive, it means that the subject is above average. When z-score is negative, it means that the subject is below average.

Confidence Interval Both central tendency and variability were point estimate as we estimate with a single number. However, we will never be sure that our estimates will accurately reflect values in the population. To deal with the uncertainty, we can create boundaries that we think the population mean will fall between, Confidence Interval.

Confidence Interval Confidence interval uses confidence level in computation. Three common confidence level: – 90%: will create the smallest interval – 95% – 99%: will create the widest interval as one wants to be more confident that the interval contains a true population parameter

Confidence Interval Formula to compute a confidence interval for a mean: