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Section 2.1 Visualizing Distributions: Shape, Center and Spread To learn the basic shapes of distributions of data: Uniform, normal and skewed To describe characteristics of a shape of distribution: Symmetry, skewness, modes, outliers, gaps and clusters To describe a uniform distribution using range and frequency To estimate graphically the mean and standard deviation of a normal distribution and use them to describe the distribution. To estimate graphically the median and quartiles and use to describe a skewed distribution. A graph that shows: Spread of the data How many times a value in the data occurs How have we used a distribution? To see where data from a simulation lies. To explore probabilities of a random selection Uniform (Rectangular) All values occur equally often Selecting the last digit of the numbers in a phone book Selecting the last digit of social security #s or you student id #s randInt(start,end,n) ie: randInt(0,9,100) L1 Why? ▪ All digits 0-9 would be used and there would be no reason any one of them would be used more than the others. Normal Distributions (bell-shaped) Very common in our world and will be used throughout the year. Measure a ball. Measure the diameter to the nearest mm and record your result. As a class create a dot plot that shows the distribution of our measurements. What do you notice? Why do you think that occurred. Normal Distribution Video #1 Normal Distribution Video #2 Characteristics of a Normal Distribution: Symmetric: The mean (avg. value) of the data is the center point. If it is truly normal, the mode and median of the data is also at the center. These are called measures of center. Standard deviation (SD) is a measure of the spread of a normal distribution. The SD happens to be the distance from the center out to the inflection point on the curve. One SD out from the center in both directions will give boundaries for an area of 68% of the total under the curve. This is a measure of the spread of the data. Skewed distribution (a longer tail on one side) Skewed right: tail stretches to the right Not a line of symmetry Median is typically used to describe a measure of center since there is not line of symmetry. ▪ Divide the plot into equal #s of data points on each side of the median. Quartiles are a measure of spread for this. ▪ Lower quartile divides the lower half of the data ▪ Upper quartile divides the upper half of the data Bimodal Distributions (two peaks) Cases often represents two groups when this occurs: Male/Female, Majority/Minority… Outliers: A data value that stands apart from the bulk of the data. These deserve special attention Sometimes they are mistakes Sometimes there are unusual circumstances that can be important to great discoveries. Gaps in where the data values lie. Could also call the areas where the bulks of the data lie, clusters. When describing a distribution, you must include the following: Shape (as we have just described) Measure of center (or centers if bimodal) ▪ mean, mode, median Measure of spread Locations of Gaps or Clusters Discussion D4 Practice P1-3, 5 Page 39 E1, 3, 5, 8, 11 AP only: also E4, and 6