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# Chapter 1 MAT 221 MAT 221 - M200

Syracuse

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This 3 page Class Notes was uploaded by Niki Neidhart on Sunday January 31, 2016. The Class Notes belongs to MAT 221 - M200 at Syracuse University taught by X. Au in Spring 2016. Since its upload, it has received 18 views. For similar materials see Elements of Mathematical Statistics and Probability Theory in Math at Syracuse University.

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Date Created: 01/31/16

1.2 – Displaying Distribution w/ Graph Categorical Data: - Bar Graph - Pie Chart *Be aware of misleading graphs (scaling) - To deemphasize, zoom out - To emphasize, zoom in - Don’t make 3D graphs Quantitative Data: - Stem plots - Histograms Outliers – observations (numbers) that lie outside the overall pattern of the distribution Symmetric – you could draw a line down the middle and both sides look the same Skewed Right – the right side of the histogram extends much farther than the left - The outliers are on the right, the majority is on the left Skewed Left – the left side of the histogram extends further than the right - The outliers are on left, the majority on the right 1.3 Describing Distributions with Numbers Measures of Center: 1. Mean or Average – to calculate, add all numbers and divide by the amount of numbers (cases) 2. Median – the midpoint of the distribution (put in order first. if no middle, take the average of the two) Ex. 0, 1, 2, 3, 100 Median – 2 Mean – 53 *Median is resistant (not effected) to outliers * Mean is “center of gravity” Symmetric – Mean = median Skewed Right – Mean > Median Skewed Left – Mean < Median Measure of Spread: Quartiles 1. First Quartile – the first 25% of the data, the median of the lower half of data 2. Third Quartile – the last 25% of the date, the median of the higher half of data • Exclude the median Interquartile Range (IQR) – the distance between quartiles IQR = Q 3 – Q 1 Min Q1 Median Q3 Max Five-number Summary – min, Q1, median, Q3, max Boxplot – above number line Rule of Thumb for Identifying Outliers: Any number lower/higher than 1.5 X IQR Standard Deviation – measures how far the observations are from their mean *affected by skewness or outliers 1. calculate mean (average) 2. s2= (x 1-x)2 + (x 2 – x)2 + … n-1 ex) Find SD for: 7,4,4,12,18 mean = 9 s2 = (7-9) ^2 + (4-9)^2 + (4-9) ^2 + (12-9) ^2 + (18-9) ^2 5-1 SD = 6 - SD is 0 when all the numbers have the same value, otherwise its positive - SD has the same unit of measurement as the original observations 1.4 Density Curves and Normal Distributions Density Curve – a smooth approximation of a histogram - Estimation, don’t have to draw histogram, just red curve - The total area under the curve is equal to 1 or 100% Density Histogram Mean = µ mean = x SD = σ SD = s - The median of a density curve is the point that divides the area under the curve in half - the mean is the point at which the curve would balance is made of solid material Normal Distributions – a special symmetrical bell shaped distribution whose density curve is completely determined by its mean (µ) or SD (σ) The 68-95-99.7% Rule for Normal Distributions: 68% • 68% are within 1 SD of the mean • 95% are within 2 SD of mean 95% • 99.7% are within 3 SD of mean 99.7% ex) How bad is your car? mean = 24.8 SD = 6.2 Gas mileage = 18.6 Z-score: the number of standard deviations that x is from the mean µ Z= x-µ/ σ *when x> the mean, z is positive *when x< the mean, z is negative

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