Statistics Chapter 3

Stem-and-Leaf Display: How to Construct

1 .Select one or more leading digits for the stem values. The trailing digits (sometimes just the first one of the trailing digits) becomes the leaves. 2. List all possible stem values in a vertical column and then draw a vertical line to the right of this column. 3. Record the leaf for every observation beside the corresponding stem value (place them in numerical order extending out from the stem from smallest to largest). 4. Indicate the units for stems and leaves someplace in the display

Pie Chart

A categorical data set can also be summarized using a pie chart. A circle is used to represent the whole data set, with "slices" of the pie representing the possible categories. To help us construct these pie charts, we should find the relative frequencies of each of the categories and have them in percentage form - Multiply by 100.

Unimodal and Symmetric

A common name for a unimodal and symmetric histogram is a Bell-shaped curve also known as a Normal model.

Cumulative Relative Frequency

A cumulative relative frequency plot is just a graph of the cumulative relative frequencies against the single value or the upper endpoint of the corresponding interval.

Bimodal

A histogram with 2 main peaks is said to be bimodal. This refers to a histogram in which 2 classes with the largest frequencies (may have slightly different values) are separated by at least one class.

Multimodal

A histogram with more than 2 peaks is said to be multimodal.

Uniform

A histogram with no peaks is said to be uniform.

Unimodal

A histogram with one main peak is said to be unimodal.

Segmented Bar Graph

A pie chart can be difficult to construct by hand. The segmented bar graph uses a rectangular bar to represent the entire data set. The bar is divided into segments (just like the "slices" of a pie chart) with different segments representing different categories. Just like the pie chart, we should find the relative frequencies (decimal form) for each category. A helpful way to find the bar segment to each category is to find a cumulative relative frequency.

Stem-and-Leaf Display

A stem-and-leaf display is an effective and compact way to summarize univariate numerical data. Each number in the data set is broken into 2 pieces, a stem and a leaf. The stem is the first part of the number that consists of the beginning digit(s). The leaf is the last part of the number and consists of the final digit(s).

Comparative Stem-and-Leaf Display

At times, a researcher would like to see whether 2 groups of data differ in some way. A comparative stem-and-leaf display can provide a visual aid to look at the comparison of the 2 groups under the same measurement. To display this, 1 group will have its leaves to the right of the stem value and the other group will have its leaves to the left.

Histograms

Even though a stem-and-leaf display can graphically display numerical data, it is ineffective when the data set contains a large number of observations. A histogram works well for large data sets.

Frequency Distributions and Histograms for Continuous Numerical Data

For continuous data, it may be difficult to display the intervals. However, we can display them in something called class intervals. A class interval looks something like 15 to <20, 20 to <25, etc. so we can distinguish where the values of 20, 25, etc. would be placed.

Grouping

If the range of the numerical values are small, then you can list every value on the histogram. However, if the range is very large (or even small), then it may be best to group the values into intervals.

2nd way drawing a bar chart with an interval of discrete values

When you have an interval of values (say from 100 - 199), then have the base of the rectangle consist of the value of 99.5 - 199.5. This new range of values is known as a class boundary (explained on the next slide). The reason behind a class boundary (with 0.5 leniency from both sides of the interval) is because we want the low value of the next class boundary to meet with the high value of the preceding one. The next interval of values would be 200 - 299 and its corresponding class boundary is 199.5 - 299.5. Hence, the upper boundary value from the first boundary meets with the lower boundary value from the second boundary.

What to look for when drawing a stem-and-leaf display

The display conveys information about a typical value in the data set the extent of spread about a typical value the presence of any gaps in the data the extent of symmetry (or skewness) in the distribution of values the number and location of peaks any outliers - an unusually small or large data value

How to construct a histogram for discrete numerical data

The instructions of constructing a histogram is almost like constructing a bar graph. Draw a horizontal scale, and mark the possible values (even if it's an interval) of the variable Draw a vertical scale, and mark it either frequency or relative frequency. There are 2 possible ways of drawing the rectangle. One way: Above each possible single value, draw a rectangle centered at that value (the rectangle for 1 is centered at 1). The height of each rectangle is determined by the corresponding frequency or relative frequency.

Symmetry

The most common histograms seen are unimodal. They come in different shapes and sizes. A unimodal histogram is symmetric if there is a vertical line of symmetry (left side is a mirror image of the right side). If it is not symmetric then it is said to be skewed. A histogram is said to be skewed to the left (negatively skewed) if the left tail of the histogram stretched out farther than the right tail and vice versa (positively skewed).

What to look for when drawing a histogram

The things we want to look for in a histogram is the same for a stem-and-leaf display. Center or typical value Extent of spread or variability General Shape (symmetric or skewed) Location and number of peaks (addressed on the next slide) Presence of gaps and outliers

Class Boundaries

The way of finding class boundaries for a single value or an interval of values of discrete numerical variables is by having one additional decimal place value and end in a 5. Examples.: A single value of 10 will have a class boundary of 9.5 - 10.5. An interval of values of 40 - 59 will have a class boundary of 39.5 - 59.5. An interval of values of 38.5 - 42.5 will have a class boundary of 38.45 - 42.55.