Lecture 5 Understanding and Comparing Distributions

Transcription

1 Lecture 5 Understanding and Comparing Distributions 1 Recall the 5-summary from our Tim Horton s example: Calories of 30 donuts. min=180, max=400, median=250, Q1=210, Q3=280 Below is the boxplot for calories of donuts: StatsCrunch: Graph>boxplot>Select Column(s)>Calories Select: use fences to identify outliers

2 2 OR, you can go back and edit the image in StatCrunch and Select the option: draw boxes horizontally. Shape: slightly right skewed Centre: the median is almost in the middle of the box. Spread (IQR): most of the donuts Calories are from 210 (Q1) to 280 (Q3). Extreme value(s): One point is plotted Individually at 400 (one donut with 400 calories we will check in chapter 6 if this 400 value is an unusual observation or not).

3 3 Steps to draw a boxplot (we use our example of Tim Horton s donut calories): Locate the min and the max on a horizontal line Locate the median, draw a vertical line above the median (away from the horizontal line). The median is inside the box of the boxplot. Locate Q1, and Q3, and draw vertical lines above these values to complete the rectangles. Calculate inner fences: o IQR = Q3-Q1 = = 70 o Lower inner fence = Q1 1.5(IQR) = (70) = 105 o Upper inner fence = Q (IQR) = (70) = 385 o Draw lines (whiskers) connecting the box to the most extreme value within fences: From Q1 draw a line to a value 105 in the data (in our example it will be 180) From Q3 draw a line to a value 385 in the data (in our example it will be 340) o Plot values outside fences individually. These points are suspected outliers (unusual observation). They are also known as extreme values.

4 Here is a display for comparing groups of donuts: Yeast, filled, and cake. This display is for 29 observations since there is only one donut that belongs to an other category, I removed it from the 30 donuts data file (we won t have a boxplot for one observation!). We describe the distributions and compare the groups: 4 Cake: Shape: about symmetric Centre: the median is in almost in the middle of the box of boxplot Spread: the width of the box (IQR) is small, from 250 to 280, which contains 50% of the Cake donuts Potential outliers: one point is plotted individually at 340. Filled: Shape: Right skewed There is no upper whisker in the filled boxplot (see stemplot, in the next page) Centre: the median is not almost in the middle of the box of boxplot Spread: the width of the box (IQR) is wider than Cake, from 210 to 270, and contains 50% of the filled donuts Potential outliers: one point is plotted individually at 400. Yeast: Shape: Right skewed Centre: the median is in not almost in the middle of the box of boxplot, and it is the lowest of all three types (200 calories) Spread: the width of the box (IQR) is the widest among the three types of donuts, from 190 to 270, which contains 50% of the yeast donuts Potential outliers: no unusual points plotted inidividually StatsCrunch: Graph>boxplot>Select Column(s)>Calories Group by: Type of donuts Select: use fences to identify outliers

5 5 *** We can have a point plotted individually, but it does not necessarily change the shape (overall pattern) of the data to be skewed. Here is the summary statistics for each group of donuts. Compare the mean with the median to determine if a distribution is symmetric or skewed. Summary statistics for Calories: Group by: Type of Donut Type of Donut n Mean Std. dev. Min Q1 Median Q3 Max IQR Cake Filled Yeast StatsCrunch: stat>summary stat>select Column(s)>Calories Group by: Type of donuts Select any statistics you d like from the drop down menu The stemplots for types of donuts: Variable: Calories for Type of Donut = Cake 2 : : 04 Variable: Calories for Type of Donut = Filled 2 : : 4 : 0 Variable: Calories for Type of Donut = Yeast 1 : : : 0 Look at all three information for the filled distribution: the boxplot, summary statistics chart, and the stemplot. We see that in the boxplot for filled, there is no upper whisker. The summary statistics chart tells us, more accurately what the value for the Q3 is, which is 270, and the max is 400. We can see in the boxplot and the stemplot, there is no value between, 270 to 400. The upper inner fence is: Q IQR = (60) = 360. This means that we would attempt to draw the upper whisker for filled from Q3 of 270 to a number in the data that is close to (less than or equal to 360. However, we do not have such value in the data, and therefore, we do not have an upper whisker. The last value in this group, 400, is plotted individually and it is beyond the upper inner fence.