1.2 N Displaying Quantitative Data with Graphs (dot plots, stemplots, histograms, shape) Target: I can graph quantitative data using dotplots and stemplots. I can describe distributions using shape, center, spread and outliers. I can describe distributions using shape, center, spread and outliers. h.w: pg 42: 37, 40, 41, 44, 45

Displaying Quantitative Variables Quantitative Variables: take numerical value for which it makes sense to find an average. Dotplots and Stemplots Dotplots are a good way to display small data sets. Dotplots are a good way to display small data sets. Stemplots are better for larger data sets (medium to large). Stemplots are better for larger data sets (medium to large).

Dotplots O ne of simplest types of graphs to display quantitative data Remember: label axis title graph scale axis

Stem (and leaf) plot (directions p. 33) Graphs quantitative data and is used for small or medium sized data. (Very common graph). Ex. 3 Watch the Caffeine (See data on p.13, don’t enter) Tip for organizing the data: First enter data into list. Then sort. Tip for organizing the data: First enter data into list. Then sort.

How to construct a stemplot 1. Separate each observation into a stem consisting of all but the rightmost digit (1 digit leafs). 2. Write stems vertically in increasing order and draw a vertical line to their right; write each leaf to right of its stem. 3. If data not entered into calculator list, write stems again and rearrange leaves in increasing order out from stem (don’t need to do if you sort with calculator).

How to construct a stemplot cont. 4. Title graph and add key describing what stems and leaves represent Key: 3/5 means the soft drink contains 35 mg of caffeine per 8 ounce serving.

Split-stem and leaf plot When splitting stems, be sure each stem is assigned an equal number of possible leaf digits. When splitting stems, be sure each stem is assigned an equal number of possible leaf digits. Notice graph “a” is very “skyscraper-ish”. Notice graph “a” is very “skyscraper-ish”. So split-stem. So split-stem.

Back-to-back stemplots - good to compare two sets of data Remember: 5 stems is a good minimum Advantages: easy to construct; display actual data values Advantages: easy to construct; display actual data values Disadvantages: does not work well with large data sets Disadvantages: does not work well with large data sets

Example: Who’s Taller? Which gender is taller, males or females? A sample of 14-year-olds from the United Kingdom was randomly selected using the Census At School website. Which gender is taller, males or females? A sample of 14-year-olds from the United Kingdom was randomly selected using the Census At School website. Here are the heights of the students (in cm): Here are the heights of the students (in cm): Construct stemplots. Report back. Construct stemplots. Report back. Male: 154, 157, 187, 163, 167, 159, 169, 162, 176, 177, 151, 175, 174, 165, 165, 183, 180 Male: 154, 157, 187, 163, 167, 159, 169, 162, 176, 177, 151, 175, 174, 165, 165, 183, 180 Female: 160, 169, 152, 167, 164, 163, 160, 163, 169, 157, 158, 153, 161, 165, 165, 159, 168, 153, 166, 158, 158, 166 Female: 160, 169, 152, 167, 164, 163, 160, 163, 169, 157, 158, 153, 161, 165, 165, 159, 168, 153, 166, 158, 158, 166

Describing the Overall Pattern of a Distribution See if you can describe the shape of the dist. in a few words. (long tail to the right/left, skewed, unimodal, bimodal-2peaked) See if you can describe the shape of the dist. in a few words. (long tail to the right/left, skewed, unimodal, bimodal-2peaked) Identify outliers: indiv. observations that fall outside the overall pattern of the graph. Identify outliers: indiv. observations that fall outside the overall pattern of the graph. Give center and spread. Give center and spread. Don’t forget your SOCS!

Patterns of Distributions Center - A value that divides the observations so that about half take larger values and about half have smaller values. Median (M) - The midpoint of a distribution, the # such that half the observations are smaller and the other half are larger.

To find the median of a distribution: 1. Arrange all observations from smallest to largest. 2. If n is odd, M is the center observation in the ordered list. 2, 3, 4 2, 3, 3, 4, 5 (n + 1)/2 th observation is the median

3. If n is even, M (median) is the mean of the two center observations in the ordered list. 1, 2, 3, 5; 10, 15, 21, 23, 30, 40 M = 2.5M = 22 Which observations if list is long? Take mean of the (n/2) th observation and the next observation.

We will usually find median with the calculator. Enter data into L1; STAT:EDIT Enter data into L1; STAT:EDIT Highlight L1 and press clear if needed. Highlight L1 and press clear if needed. Enter data. Ex. 1,1,2,4,6,7,9,10 Enter data. Ex. 1,1,2,4,6,7,9,10 2 nd STAT: MATH: median(L1): enter 2 nd STAT: MATH: median(L1): enter

Spread - Can be described by giving smallest and largest values.

Shape - Shape of graph of distribution. Symmetric - Right and left sides of histogram are approximate mirror images of each other.

Skewed to the right (positively skewed) Right side of histogram extends much farther out than left side Right side of histogram extends much farther out than left side pulled to the right pulled to the right long tail to the right long tail to the right

Skewed to the left (negatively skewed) Left side of histogram extends much farther out than right side; pulled to the left, long tail to the left. (Draw)

Ex 4.Michigan College Tuitions There are 81 colleges and universities in Michigan. Their tuition and fees for the 1999 to 2000 school year run from $1260 at Kalamazoo Valley Community College to $19,258 at Kalamazoo College. Complete and report back (4 min).

Ex 4. Michigan College Tuitions cont. a. What do the stems and leaves represent in the stemplot? ($1260 to $19,258 ) Stems = thousands Leaves = hundreds Have the data been rounded? The data have been rounded to the nearest $100. Note: stems can be multiple digits, leaves can only be the right most digit.

b. Describe the shape, center, and spread. Shape: The distribution is skewed strongly to the right with a peak at the one stem. Center: 45 (approx. $4500) Spread: approx. 19,000 ($1300 to $19300) approx. 19,000 ($1300 to $19300) Median is 41 st value – count them.

Are there any outliers? The observations 182 and 193 appear to be outliers. ($18,200, $19,300) The observations 182 and 193 appear to be outliers. ($18,200, $19,300)

Displaying Quantitative Data Comparing Distributions Some of the most interesting statistics questions involve comparing two or more groups. – Always discuss shape, center, spread, and possible outliers whenever you compare distributions of a quantitative variable. Compare the distributions of household size for these two countries. Don’t forget your SOCS! Don’t discussing SOCS for each distribution separately. You will not receive credit! Use phrases like “about the same as” or “is much greater than” to receive full credit. Place U.K South Africa

Think – Pair - Share In small group, read over comparing distributions example on page 32. In small group, read over comparing distributions example on page 32. Do problem #43 on page 43. Do problem #43 on page 43. Report out. Report out.