Statistics 200 Lecture #4 Thursday, September 1, 2016 Textbook: Sections 2.6 through 2.7 Objectives (relating to quantitative variables): • Use side-by-side boxplots to explore relationship between quantitative and categorical variable • Understand intuition for two basic measures of spread: – IQR (interquartile range) – Standard deviation • Learn and apply empirical rule for bell-shaped (normal) distributions

Yesterday: Descriptive Methods for Quantitative Data Graphs: histogram, boxplot numbers: statistics measures of center: mean, median

Sex Side by side boxplots: Compare the distributions of two or more groups Include an explanatory variable:________ Sex

Side by side boxplots: Compare the distributions of two or more groups

Example 3: Comparing Two or More Boxplots - What to look for Part of Graph Comparisons Conclusions within box compare ______ (center) smallest: largest: compare spread (IQR) compare shapes slightly right skewed: left skewed: outside box compare number of outliers least: most: Median Female Male Female Male M Female Male Female

Measures of Spread (Variation) Standard Deviation Interquartile Range (IQR) Resistant Sensitive __________ measure _________ measure IQR = Q3 - Q1 Range: Max - Min

First : Range and IQR = 135 – 45 = 90 = 100-90 = 10 Range = max – min IQR = Q3 - Q1 = 135 – 45 = 90 = 100-90 = 10 Min Q1 Median Q3 Max 45 90 95 100 135

First : Range and IQR entire 90 50% middle 10 Range Interpretation: amount of variation in the_______ sample is _____ mph entire 90 IQR Interpretation: amount of variation in the _________ _________ of the sample is ______ mph middle 50% 10

Standard Deviation: Formula The sample standard deviation is roughly the average distance between an observation and the sample mean. Measures the variability among observed data values.  

Sample 1 (10 10 10 10 10) no s = ____ Sample: has ____ variation s = ____ no Sample: has ____ variation Mean = 10

2.7: Bell-shaped Distributions and Standard Deviations The good stuff! A bell-shaped distribution is a special kind of symmetric distribution. We care greatly about Normal distributions. The standard deviation tells us a LOT for a normal distribution.

Normal Distribution normal The ______ distribution is one of the most commonly seen distributions for quantitative data. A normal distribution has a distinctive symmetric bell shape.

Normal Distribution: The empirical rule Also called the “68-95-99.7 rule”

Empirical Rule (E.R.) For a normal distribution… 68 _______% of the values (data) fall: within ± ______ st dev of the mean 1 ____% of the values (data) fall: within ± _____ st dev of the mean 95 2 99.7 _____% of the values (data) fall: within ± _____ st dev of the mean 3

Visualization of the Empirical Rule Notice: The middle 95% has a width of 4 standard deviations.

(for roughly bell-shaped distributions) Rough way to approximate the standard deviation (for roughly bell-shaped distributions) Based on the empirical rule: • Look at the histogram and estimate the range of the middle 95% of the data. • The standard deviation is about one fourth of this range.

What is the best estimate below of the s.d. for these data? 10 20 30 40 50

Example Survey Question: “What is your height in inches? STAT 200 survey results (SP 2016) found: a normal shape mean is 67 inches & st dev is 3 inches. Range of values 3.7, 4.0, 4.3, 4.6, 4.9, 5.2, 5.5

Exercise: Sketch distribution of Actual Heights Mean = 67 inches; Exercise: Sketch distribution of Actual Heights Mean = 67 inches; StDev = 3 inches

Visualize heights distribution with histogram s = 3 inches

Answer Questions using histogram What percent of Stat 200 students are at most 64 inches in inches in height? Answer:

Answer Questions using histogram Which percentile is located 2 standard deviations below the mean? Answer:

Answer Questions using histogram Which height represents the 84th percentile? A. 61 inches B. 64 inches C. 67 inches D. 70 inches E. 73 inches

Review: If you understood today’s lecture, you should be able to solve 2.61, 2.87, 2.95, 2.101, 2.103, 2.105, 2.109abc Recall objectives: • Use side-by-side boxplots to explore relationship between quantitative and categorical variable • Understand intuition for two basic measures of spread: – IQR (interquartile range) – Standard deviation • Learn and apply empirical rule for bell-shaped (normal) distributions