Chapter 4 Review December 19, 2011

Box Plots 5 points are graphed (minimum, 1st quartile, median, 3rd quartile, and maximum). Box goes around 1st quartile median and 3rd quartile. Line goes vertically through median. Lines extend horizontally from box to minimum and maximum, unless there are outliers. If there is an outlier, the line extends to the closest number to the outlier, and the outlier has a star, x or point to represent it. Can be shown vertically or horizontally.

Stem Plots Stem is made up of all digits of number except for last digit. Leaf is made up of last digit. All repeats are listed. All numbers are listed in order, smallest to largest. Even if a number isn’t used on the stem, you put it there (i.e., there are 20’s, 30’s, 50’s, but no 40’s…you still put the 4). There is a vertical line dividing the stem and the leaves. There is to be a key to demonstrate how to read the number (ex., 2|3 = 23)

Dot Plots There is a horizontal number line on this graph. Dots are placed vertically to represent all data. Graph can show trends.

Frequency Histogram Vertical axis shows count of data which meet criteria. Horizontal axis is grouped in ranges of data (i.e., 0-4, 5-9, etc.). All data must only fit into one category. Frequency histograms are good for quantitative data only.

Relative Frequency Histogram Vertical Axis shows percentage or fraction of data which meet criteria. Horizontal axis is grouped in ranges of data (i.e., 0-4, 5-9, etc.). All data must only fit into one category. Relative frequency histograms are good for quantitative data only. Has the same shape as a frequency histogram.

Shapes of Histograms Symmetric—roughly bell shaped on both sides. Uniform—all bars roughly the same height. Skewed right—most of data is on left, tail is on right. Skewed left—most of data is on right, tail is on left.

Mean The average of the data. Represented by x-bar. If you are talking about a symmetric graph, it’s the center of the graph.

Standard Deviation Represents the average distance the data is from the mean. Step 1: Find x-bar. Step 2: Find the deviations of each piece of data (x minus x-bar). Step 3: Find the squared deviations of each piece of data (x minus x-bar) squared. Step 4: Add up all of the values from Step 3.

Standard Deviation Step 5: Take Step 4’s value and divide by n-1 (n is the number of values in the data set). This is the variance. Step 6: Take the square root of Step 5’s value. That’s the standard deviation.

Five Number Summary Minimum – Smallest number on list. Quartile 1 – Middle number between minimum and median of list. Median – Middle number of list (after list has been organized smallest to largest). Quartile 3 – Middle number between median and maximum. Maximum – Largest number on list.

5 Number Summary Examples 1, 2, 3, 4, 5, 6, 7, 8, 9 Minimum = 1, Median = 5, Maximum = 9 When calculating Q1, the median is not included (already used), so you are only looking at 1, 2, 3, 4 Quartile 1 = 2.5 When calculating Q3, the median is not included (already used), so you are only looking at 6, 7, 8, 9 Quartile 3 = 7.5

5 Number Summary Examples 1, 2, 3, 4, 5, 6, 7, 8 Minimum = 1, Median = 4.5, Maximum = 8 Median is the average of the two middle numbers when there is an even number of terms. Quartile 1 data consists of 1, 2, 3, 4 (because 4 is not the median). Quartile 1 = 2.5 Quartile 3 data consists of 5, 6, 7, 8 (because 5 is not the median). Quartile 3 = 6.5

Inter Quartile Range This is the value Quartile 3 – Quartile 1. Can be used to determine if outliers exist.

Calculating Outliers If it exists, there will only be one outlier at one or both ends of a list of data. To calculate if a number is an outlier, multiply 1.5(Q3-Q1). Find the distance Maximum – Q3 (or Q1 – Minimum). If the distance is larger than the 1.5(Q3-Q1) calculation, then that number is an outlier. If the distance is smaller, the number is not an outlier.

Standard Deviation and Mean vs. 5 Number Summary Best to use standard deviation and mean to describe data when histogram is symmetric and free of outliers. Best to use 5 number summary to describe data when histogram is skewed and/or has outliers.

Other Graphs Ogives – Cumulative Frequency Graph Frequency Chart – Tally Chart

Variables Categorical – also known as qualitative. Usually non-numeric. Quantitative. Always numeric, can do math on data.