Introductory Statistics

Box Plots Math 140 Introductory Statistics „ Example: Mammal speeds, „ 11,12,20,25,30,30,30,32,35, 39,40,40,40,42,45,48,50,70. „ „ „ „ „ „ „ Modified Box Plots „ Example: Mammal speeds, „ A Modified Box Plot also 11,12,20,25,30,30,30,32,35, 39,40,40,40,42,45,48,50,70. min Q1 Q2 Q3 30 42 IQR Range max takes into account the outliers. „ An outlier is a value which is more than 1.5 times the IQR from the nearest quartile. „ Example: „ (1.5)IQR = 18 „ 11 < 30 – 18, so 11 is an outlier. „ 70 > 42 + 18, so 70 is an outlier. min = 11 Q1 = 30 Median = 37 Q3 = 42 max = 70. A Box Plot is a graphical display of a five-point summary. min Q1 Q2 Q3 max IQR Range = 70 – 11 = 59 IQR = 42 – 30 = 12 Range Modified Box Plots „ Example: Mammal speeds, 11,12,20,25,30,30,30,32,35,39 ,40,40,40,42,45,48,50,70. min Q1 Q2 Q3 „ Example: „ „ „ max (1.5)IQR = 18 11 < 30 – 18, so 11 is an outlier. 70 > 42 + 18, so 70 is an outlier. IQR Range 1 Modified Box Plots „ Example: Mammal speeds, Q1 Q2 Q3 „ Box Plots and Modified Box Plots are useful „ Example: 11,12,20,25,30,30,30,32,35,39 ,40,40,40,42,45,48,50,70. min Box Plots (Modified) „ „ „ max (1.5)IQR = 18 11 < 30 – 18, so 11 is an outlier. 70 > 42 + 18, so 70 is an outlier. when plotting a single quantitative variable and: want to compare shape, center, and spread of two or more distributions. „ The distribution has a large number of values „ Individual values do not need to be identified. „ (Modified) We want to identify outliers. „ Q1 Q2 Q3 IQR Range Section 2.4 Recentering and Rescaling „ Recentering a data set (adding the same number c to all the values in the set) „ „ Shape or spread do not change. It slides the entire distribution by the amount c, adding c to the median and the mean. „ Rescaling a data set (multiplying all the values in the set by the same positive number d) „ „ Basic shape doesn’t change. It stretches or shrinks the distribution, multiplying the spread (IQR or standard deviation) by d and multiplying the center (median or mean) by d. Example City Country Temperature (oF) Addis Ababa Ethiopia 32 Algiers Algeria 32 Bangkok Thailand 50 Madrid Spain 14 Nairobi Kenya 41 Brazilia Brazil 32 Warsaw Poland -22 2 The Influence of Outliers Example „ A summary statistic is resistant to outliers if it does not change very much when an outlier is removed. „ sensitive to outliers if the summary statistic is greatly affected by the removal of outliers. „ Percentiles and CRF plots 2.5 The Normal Distribution „ You are responsible to read through this and „ Shape understand the concepts of percentile, and cumulative relative frequency plot. SD SD Mean „ Center: Mean x= sum of values ∑x = n number of values „ Spread: Standard Deviation σ n −1 = ∑ ( x − x) n −1 2 3 Applications of the Normal Distribution The Two Main Problems. „ The normal distribution tells us how: „ Variability in measures behaves. „ Variability in population behaves. „ Averages and some other summary statistics behave when you repeat a random process. „ „ Nice property: A normal distribution is determined by its mean and standard deviation! (If you know mean and SD you know everything) Unknown percentage problem. „ The distribution of the SAT scores for the University of Washington was roughly normal in shape, with mean 1055 and standard deviation 200. 1. What percentage of scores were 920 or below? The distribution of the SAT scores for the University of Washington was roughly normal in shape, with mean 1055 and standard deviation 200. 1. What percentage of scores were 920 or below? (Unknown percentage problem) 2. What SAT score separates the lowest 25% of the SAT scores from the rest? (Unknown value problem) Unknown value problem. „ The distribution of the SAT scores for the University of Washington was roughly normal in shape, with mean 1055 and standard deviation 200. 2. What SAT score separates the lowest 25% of the SAT scores from the rest? 4 Which one is it? 1. Unknown percentage problem. Given x, Find P. The Standard Normal Distribution. „ It is the normal distribution with Mean = 0, and standard deviation = 1. The area under the curve equals 1 (or 100%) 2. Unknown value problem. Given P, Find x. -1 The Standard Normal Distribution. „ It is the normal distribution with Mean = 0, and standard deviation = 1. The area under the curve equals 1 (or 100%) „ The Standard Normal Distribution is important because any normal distribution can be recentered and/or rescaled to the standard normal distribution. This process is called standarizing or converting to standard units. „ Also, the two main problems can be easily solved in the Standard Normal Distribution with the help of tables or a calculator. SD SD 0 1 The Two Main Problems in the Standard Normal Distribution. Unknown Percentage. (Given z, Find P ) „ With Table A. Last page in the book „ Use the units and the first decimal to locate the row and the closest hundredths digits to locate the column. The number found is the percentage of the number of values below z. „ With Calculator „ Enter normalcdf(-99999, z) to get the percentage of the number of values below z. 5 Example 6