Chapter 1: Exploring Data Sec. 1.2: Displaying Quantitative Data with Graphs

Learning Objectives After this section, you should be able to: The Practice of Statistics, 5 th Edition2 MAKE and INTERPRET dotplots and stemplots of quantitative data DESCRIBE the overall pattern of a distribution and IDENTIFY any outliers IDENTIFY the shape of a distribution MAKE and INTERPRET histograms of quantitative data COMPARE distributions of quantitative data Displaying Quantitative Data with Graphs

Dotplots Dotplot - Each data value is shown as a dot above its location on a number line. How to make a dotplot: 1) Draw a horizontal axis (a number line) and label it with the variable name. 2) Scale the axis from the minimum to the maximum value. 3) Mark a dot above the location on the horizontal axis corresponding to each data value. How to make a dotplot: 1) Draw a horizontal axis (a number line) and label it with the variable name. 2) Scale the axis from the minimum to the maximum value. 3) Mark a dot above the location on the horizontal axis corresponding to each data value. Quantitative data Values at the bottom Each dot = 1 piece of data Heights are how many at each value

Number of Goals Scored Per Game by the 2012 US Women’s Soccer Team Dotplots Example, p. 25

Stemplots Stemplots (a.k.a. stem-and-leaf plot) give us a quick picture of the distribution while including the actual numerical values. How to make a stemplot: 1)Separate each observation into a stem (all but the final digit) and a leaf (the final digit). 2)Write all possible stems from the smallest to the largest in a vertical column and draw a vertical line to the right of the column. 3)Write each leaf in the row to the right of its stem. 4)Arrange the leaves in increasing order out from the stem. 5)Provide a key that explains in context what the stems and leaves represent. How to make a stemplot: 1)Separate each observation into a stem (all but the final digit) and a leaf (the final digit). 2)Write all possible stems from the smallest to the largest in a vertical column and draw a vertical line to the right of the column. 3)Write each leaf in the row to the right of its stem. 4)Arrange the leaves in increasing order out from the stem. 5)Provide a key that explains in context what the stems and leaves represent. Quantitative data Around 5 stems is good minimum May have to split (high/low) Numbers listed in increasing order Each stem must have an equal # of possibilities May need to round data (5.3 ≈ 5) Leaves can only be one digit

Example, p. 31 These data represent the responses of 20 female AP Statistics students to the question, “How many pairs of shoes do you have?” Construct a stemplot Stems Add leaves Order leaves Add a key Key: 4|9 represents a female student who reported having 49 pairs of shoes.

When data values are “bunched up”, we can get a better picture of the distribution by splitting stems Key: 4|9 represents a student who reported having 49 pairs of shoes. Females “split stems” Example, p. 32 ***Splitting stems is also useful if you do not have the recommended minimum of 5.

Example, p. 32 Two distributions of the same quantitative variable can be compared using a back-to-back stemplot with common stems. Key: 4|9 represents a student who reported having 49 pairs of shoes Males Females Males ***Note that the distribution on the left has the leaves decreasing in order (increasing going out from the stem)

To understand a graph ask, “What do I see?” Examining the Distribution of a Quantitative Variable How to Examine the Distribution of a Quantitative Variable S hape – Look at main features; look for rough symmetry or clear skewness. Use modifiers!!!! (roughly, approximately, somewhat, etc.) O utliers – outcomes that deviate from the overall pattern. C enter - where the middle of the distribution falls. S pread – (for right now) smallest to largest value. How to Examine the Distribution of a Quantitative Variable S hape – Look at main features; look for rough symmetry or clear skewness. Use modifiers!!!! (roughly, approximately, somewhat, etc.) O utliers – outcomes that deviate from the overall pattern. C enter - where the middle of the distribution falls. S pread – (for right now) smallest to largest value. ***We will discuss Center & Spread more in Sec. 1.3.***

Describing Shape A distribution is: roughly symmetric if the right and left sides of the graph are approximately mirror images of each other. A distribution is: roughly symmetric if the right and left sides of the graph are approximately mirror images of each other

Describing Shape skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side

Describing Shape skewed to the left (left-skewed) if the left side of the graph is much longer than the right side.

Remember … Which way is the skier skiing? Which side is the tail on?

Shape? Symmetric Skewed-left Skewed-right

Other Shapes UnimodalBimodalUniform

Comparing Distributions Always discuss shape, center, spread, and possible outliers whenever you compare distributions of a quantitative variable. Not enough to just say what the center and spread are…compare them…Distribution A has a center that is larger than Distribution B.

Example, p. 30 Compare the distributions of household size for these two countries. Don’t forget your SOCS! Shape: The distribution of household sizes in South Africa is skewed right, while the distribution of household sizes in the U.K. is roughly symmetric.

Example, p. 30 Compare the distributions of household size for these two countries. Don’t forget your SOCS! Outliers: South Africa appears to have 1 maybe 2 outliers at 26 & 15. The U.K. does not appear to have any outliers.

Example, p. 30 Compare the distributions of household size for these two countries. Don’t forget your SOCS! Center: The center for South African household sizes seems to be larger than the U.K.. South Africa – 6, U.K. – 4

Example, p. 30 Compare the distributions of household size for these two countries. Don’t forget your SOCS! Spread: The distribution for households in South Africa has more variability 3 to 26 people) than that of the U.K. (2 to 6).

Homework – Due Monday  P. 41 #37, 41, 45, 50