Understanding Basic Statistics Chapter 2 Organizing Data Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared by Jon Booze

Frequency Tables A frequency table organizes quantitative data. partitions data into classes (intervals). shows how many data values are in each class. Test Score Number of Students 61-70 4 71-80 8 81-90 15 91-100 7

Data Classes and Class Frequency Class: an interval of values. Example: 61  x  70 Frequency: the number of data values that fall within a class. “Five data fall within the class 61  x  70”. Relative Frequency: the proportion of data values that fall within a class. “18% of the data fall within the class 61  x  70”.

Structure of a Data Class A “data class” is basically an interval on a number line. It has: A lower limit a and an upper limit b. A width. A lower boundary and an upper boundary (integer data). A midpoint.

Structure of a Data Class A “data class” is basically an interval on a number line. If a = 60 and b = 69 for integer data, what is the value of the lower boundary? a). 60 b). 59.5 c). 9 d). 64.5

Structure of a Data Class A “data class” is basically an interval on a number line. If a = 60 and b = 69 for integer data, what is the value of the lower boundary? a). 60 b). 59.5 c). 9 d). 64.5

Constructing Data Classes Find the class width. Increase the computed value to the next higher whole number. Find the class limits. The lower limit of the “leftmost” class is set equal to the smallest value in the data set.

Constructing Data Classes, cont’d Find the class boundaries (integer data). Subtract 0.5 from the lower class limit and add 0.5 to the upper class limit. For a certain data set, the minimum value is 25 and the maximum value is 58. If you wish to partition the data into 5 classes, what would be the class width? a). 5 b). 6 c). 7 d). 8

Constructing Data Classes, cont’d Find the class boundaries (integer data). Subtract 0.5 from the lower class limit and add 0.5 to the upper class limit. For a certain data set, the minimum value is 25 and the maximum value is 58. If you wish to partition the data into 5 classes, what would be the class width? a). 5 b). 6 c). 7 d). 8

Building a Frequency Table Find the class width, class limits, and class boundaries of the data. Use Tally marks to count the data in each class. Record the frequencies (and relative frequencies if desired) on the table.

Histograms Histogram – graphical summary of a frequency table. Uses bars to plot the data classes versus the class frequencies.

Making a Histogram Make a frequency table. Place class boundaries on horizontal axis. Place frequencies on vertical axis. For each class, draw a bar with height equal to the class frequency and width equal to the class width plus 1.

Making a Histogram

Distribution Shapes Symmetric Uniform Bimodal Skewed Left Skewed Right

Critical Thinking A bimodal distribution shape might indicate that the data are from two different populations. Outliers – data values that are very different from other values in the data set. Outliers may indicate data recording errors.

Exploratory Data Analysis EDA is the process of learning about a data set by creating graphs. EDA specifically looks for patterns and trends in the data. EDA also identifies extreme values.

Graphical Displays… … represent the data. … induce the viewer to think about the substance of the graphic. …should avoid distorting the message of the data.

Bar Graphs Used for qualitative or quantitative data. Can be vertical or horizontal. Bars are uniformly spaced and have equal widths. Length/height of bars indicate counts or percentages of the variable. “Good practice” requires including titles and units and labeling axes.

Bar Graphs Example:

Pareto Charts A bar chart with two specific features: Heights of bars represent frequencies. Bars are vertical and are ordered from tallest to shortest.

Circle Graphs/Pie Charts Used for qualitative data Wedges of the circle represent proportions of the data that share a common characteristic. “Good practice” requires including a title and either wedge labels or legend.

Time-Series Shows data measurements in chronological order. Data are plotted in order of occurrence at regular intervals over a period of time.

Critical Thinking – which type of graph to use? Bar graphs are useful for quantitative or qualitative data. Pareto charts identify the frequency in decreasing order. Circle graphs display how a total is dispersed into several categories. Time-series graphs display how data change over time.

Critical Thinking – which type of graph to use? What type of graph would be best for showing the ice cream flavor preferences of a group of 100 children? a). Histogram b). Pareto graph c). Time series graph d). Circle graph

Critical Thinking – which type of graph to use? What type of graph would be best for showing the ice cream flavor preferences of a group of 100 children? a). Histogram b). Pareto graph c). Time series graph d). Circle graph

Stem and Leaf Plots Displays the distribution of the data while maintaining the actual data values. Each data value is split into a stem and a leaf.

Stem and Leaf Plot Construction

Critical Thinking By looking at the stem-and-leaf display “sideways”, we can see the distribution shape of the data.

Critical Thinking Large gaps between stems containing leaves, especially at the top or bottom, suggest the existence of outliers. Watch the outliers – are they data errors or simply unusual data values?