A z-score is directional. The absolute value of z tells you how many standard deviations the score is from the mean. The sign of z tells you whether its greater than or less than the mean. Z scores give you the ability to compare values across distributions with different means and standard deviations. 2.1: Measures of Relative Standing and Density Curves

Jenny scored an 86 on her first stats test. How did she perform among her classmates? 1) Look at distribution Outliers? Shape? 2) Summary Stats

1) Jenny scored above average. But by how much? 2) Katie scored the highest, 93. What is her z-score? What does it mean? 3) Norman got a 72. what is his z-score? What does it mean?

Percentiles Norman got a 72 on his exam. Only one person did worse than he did out of a total of 25 people. What is his percentile? Katie got the highest score out of the class (she was the 93). What is her percentile?

On an index card, write your height in inches, then write your height on the board. Hold up your index card and put yourselves in order around the room (shortest to tallest). Count the number of people who are shorter than you (include yourself). Calculate the mean, standard deviation, 5 # summary. Calculate your percentile, then find how many standard deviations you are above or below the mean (find your z-score). Write your percentile and z-score on the back of your index card, and hold it up when Ms. O. tells you to.

On Baby Leo’s one year birthday, his doctor Dr. Fred gave Ms. O. his percentiles: 99% for height 83% for weight 90% head circumference What do these mean?

You can use this inequality for any approximation (normal or skewed). Describes the percent of observations in any distribution that fall within a specified number of standard deviations of the mean.

Strategy for exploring data on a single quantitative variable: 1. Graph it (stemplot/histogram) 2. Overall pattern? Striking deviations? 3. Numerical summary to describe center/spread? 4. Describe pattern w/smooth curve if it’s regular = density curve

Density Curve Example Regular Distribution Symmetric Both tails fall off smoothly from single center peak No gaps/obvious outliers Smooth curve = good overall description of overall pattern of the data Curve is a mathematical model for the distribution (ignores irregularities and outliers)

From histogram to density curve

Why a smooth curve? Histogram depends on our choice of classes, but when we use a curve, it doesn’t depend on any choices we make (easier to work with) Use a smooth curve to describe what proportion of the observations fall in each range of values, not the counts of the observations. Our eyes respond to the areas of the bars in a histogram. Same is true of a smooth curve: areas under the curve represent proportions of the observations. We adjust the scale of the graph so the total area under the curve = 1.

Important Points…. 1. The curve doesn’t describe outliers! 2. It is an idealized description of the data – an “approximation” – but is accurate enough for practical use (no real set of data is exactly described by a density curve) 3. Foundation for probability!

Example 2.5: Reading d.c.’s Skewed slightly left Shaded area: 7-8 Area under the curve =.12 Therefore, the proportion.12 of all observations from this distribution have values between 7 and 8.

Density Curves have many shapes. Left: The median and mean of a symmetric density curve are equal. Right: The median and mean of a right-skewed density curve are not equal (mean pulled towards tail)

Since areas under a density curve represent proportions of the total # of observations… Median of a density curve is the equal areas point, the point with 50% the area under the curve to its left, and the remaining 50% of the area to the right. Quartiles divide the area under the curve into quarters (25% of the area under the curve is to the left of Q1…)

The mean is the point at which the curve would balance if it were made of solid material. The balance point! Mean of a density curve

When does Mean = Median? The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

Notation Mean and standard deviation for actual observations: and s. (“x-bar” and “s”) Mean and standard deviation for idealized distributions: and (“mew” and “sigma”)