STAT 110 1nd Edition Lecture 10 Outline of Previous Lecture I. Measures of Spread II. Mean vs. MedianIII. Standard Deviation vs. Quartiles Outline of Current LectureI. Normal DistributionII. Standard Scores (Z-scores)Current LectureI. Normal Distribution a. The proportion of observations in any given distribution is the area under the curve. The total area under the curve should equal ONE. Once a histogram is smoothed with a curve, and the area under it is equal to ONE, we call it a densitycurve. i. Describing the center of a distribution when using a density curve: the mean and median have the same interpretations as they would in a histogram. That is, the mean is the simple arithmetic average, and the median divides the data in half. ii. In a symmetric distribution, the mean is equal to the median, and they both rest at the center of the curve. On a skewed curve, the mean is pulled away from the median in the direction of the skew. If the curve is skewed left, the mean will be further to the left than the median. b. A normal distribution is unimodal (only one mode) and symmetric. It is also referred to as a “bell-shaped” curve. The distribution is described by the mean (denoted by the Greek letter mu (µ)) and the standard deviation (denoted by the Greek letter sigma (σ)). i. The mean, µ, defines the center, and the standard deviation, σ, controls the spread. The standard deviation is the distance from the mean to each data point where the curve changes direction. The larger the standard deviation gets, the more spread out the distribution is. c. The Empirical Rule (The 68-95-99.7% Rule) These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.i. 68% of the data values will fall between one standard deviation above themean and one standard deviation below the mean. 95% of the data values will fall between two standard deviations above the mean and twostandard deviations below the mean. 99.7% of the data values will fall between three standard deviations above the mean and three standard deviations below the mean. II. Standard Scores (Z-scores) a. The standard score is the number of standard deviations the data point is above or below the mean. If the data point is below the mean, the standard score will be negative. If the data point is above the mean, the standard score will be positive. b. A standard score is sometimes referred to as a z-score. Z-scores are used to standardize the distribution. When we standardize a variable, Z has a standard normal distribution with a mean of 0 and a standard deviation of 1. You can calculate a z-score for any data point using the following formula: i. Z = (x - µ)/σ where x is your specific data point, µ is the mean of the distribution, and σ is one standard deviation of the