Measures of Central Tendency
Statistical analysis often begins with description of typical values of
variables, their means, medians, and modes, also called the measures of central
tendency. The arithmetic mean is a measure of central tendency commonly
referred to as an 'average.' The median was discussed by Gauss in 1816 and the
idea was elaborated by Fechner in 1878. Fechner called the median Centralwerth,
the central value of an ordered series of scores, symbolized by the letter C.
The mode, Dichtestewerth, was defined as the locus of a distribution where it is
densest, symbolized by the letter D.
Learning Statistics by Doing Statistics
Task A
Professor Stanley administered a statistics test to seven students in his class.
The students earned the following scores: 10, 6, 6, 6, 8, 9, 7. Our first task
is to find the mean, the median, and the mode.
1. Data Entry
Start a new project by choosing (Projects,
New Project). Next, choose (Data, Enter)
to bring up the Data Entry window. Click on the default letter A and label
the first variable as X. Press the Enter key. The cursor will be advanced to
the first data cell.
Enter the following seven scores (10, 6, 6, 6, 8, 9, 7).
Remember to press the Enter key following each data entry. Note that the
cursor should be on row 8, marked with the pencil tip.
Click the
Accept button and the data set will be transferred to
the Vector Display. By default, the associated descriptors will be displayed
toward the bottom. Note that there are 7 cases (n = 7). However, only six
cases are visible. Click on Expand to show all 7 cases.
Drag the top blue
bar to move the Vector Display window to a preferred position.
2. Name the Project
Choose (Projects,
Project Name). Type Measures of Central Tendency in the
textbox. Click Accept.
3. The Median and
the Mode
A distribution means the arrangement of any set of scores in order of
magnitude. First,
arrange the scores in order from smallest to largest values Select (Modify,
Sort). Click Sort Order and the Sort in Ascending Order option will appear.
Select the variable X and click Append. The ordered
series of scores will be added and the new variable is automatically labeled as SortAsc.
Note that this variable has odd numbers of cases (n = 7)
and the median represents the
midpoint of a distribution. Thus, the median (C) is 7. Half of the other
scores are below it and half are above it. Also, notice that the score of 6
occurs three times. It is the most frequently occurring score in the
distribution of the variable. The mode, D, is 6.
4. The Mean and
the Median
Select
Descriptors: Click on any descriptors on the Vector Display to bring up optional
descriptors. Select Sum, Median, and click Accept.
The mean (M) can be defined as the sum (∑)
of all scores, divided by the number of
cases (n). Thus, M = 52/7 = 7.429.
Note that the mean is pulled higher than the median because of these three
scores (8, 9, and10).
5. Create a Frequency Table
The
frequency of a given score is the number of times the score occurs. A
frequency table can be constructed by listing scores in ascending order with
their corresponding frequencies. Choose (Frequencies, Frequencies)
from the top menu. Select the variable SortAsc and click Accept. Examine the
frequency table. The lowest value is 6 and the highest value is 10. Note that there are three students who
earned a score of 6, one student who earned a score of 7, one student who
earned a score of 8, and etc. The mode is
6.
6. Rename a
Variable Using a Shortcut
Click on any variable name
on the Vector Display to bring up the Specify Column Names dialog
box. Erase the variable name, argFreq. Rename it to Test Scores. Click
Accept.
7. Visualize of
a Frequency Distribution
Create
a line chart with the test scores plotted on the horizontal axis and the frequency
plotted on the
vertical axis. These two variables, Test Scores and
Frequency, are required to plot the line graph. Select Graphs from the Modules
bar. Click Line Graphs under the Graphs Indexed by Attributes category.
Abscissa: Select the variable Test Scores. Ordinate: select the variable
Frequency. Click Accept. Change the default 3-D graph to a 2D
graph by clicking on the 3D/2D icon. The resulting line graph would look
like this.
Note that the chart has a single peak. The distribution is not symmetric.
The tail is toward larger values. The distribution is skewed to the
right. It has a positive skew. In general, the mean will be higher
than the median when a distribution has a positive skew.
8. Label the Axes and the Chart Using Shortcuts
To label the X axis, right click a data value
on the X-axis to bring up a shortcut menu. Select Edit title and type Test Scores in the textbox.
Click
anywhere outside the textbox to exit.
To label the Y axis, right click a data value on the Y-axis to bring up a
shortcut menu. Select Edit title and type Frequency in the
textbox. To label the chart, right click
the Chart area (the gray area outside the plot) to bring up a shortcut menu and
select Edit title. Type Distribution of Test Scores in the
textbox. You may copy and paste the chart to a word
processor. Close the Chart window.
9. Produce Descriptive Statistics and Frequency Tables
Choose (Analysis I, Descriptive
Statistics). Select the variable X. Select
n, Median, Mean, List Data, and click Enter. The data set and the associated
descriptive statistics will be displayed in the output window. To obtain the frequency table,
choose (Frequencies, List Frequencies). Select the variable
X and click Accept. The frequency table will appear, along with other
related tables (cumulative frequencies and the proportions). Copy and
paste the results to a word process.
Cumulative
Frequency: The cumulative
frequency is the running total of frequencies. For example, the cumulative
frequency for the score of 7 is 3 + 1 = 4. The cumulative
frequency for the score of 9 is 3 + 1 + 1 + 1 = 6.
Relative
Frequency: Frequency counts can be measured in terms of proportions or
percentages. For example, there were three student who earned a score of 6.
The total number of students is 7. Thus, 3 / 7 = .43 = 43%. About 43 percent of the students scored 6 on the
test.
Cumulative
Proportions: For our example, approximately 86%
of the students had a score of 9 or less.
10. Compute
the Median for Even Number of Cases
Median is
computed differently for odd and even number of cases.
If the number of scores in the distribution is
even, the median is the middle value extrapolated from the adjacent scores to
the theoretical midpoint of the distribution. This extrapolation is frequently
accomplished by averaging both adjacent scores.
To create an even set of numbers, use the
Truncate command to shorten the length of the variable X.
(1) First, delete the variables on the Vector
Display except the variable X. The
three variables to be cleared are SortAsc, argFreq, and Frequency.
Choose (Data, Delete). Click the variable SortAsc and hold down
the Shift key while pressing the Down Arrow key twice to highlight the variables
we want to remove. Release the Shift key and click Clear and Compact
or Clear Selected Variables
as shown below.
(2) Truncate the variable X: Adjust the
length of the variable X to six cases. Choose (Reshape,
Truncate).
Select the variable X. Define Length: 6. Click Accept.
(3) Sort Data: A distribution means the arrangement
of any set of scores in order of magnitude. To form a distribution of the
scores, sort the scores in ascending order. Choose (Modify, Sort).
Click Sorting Order to select Sort in Ascending Order. Select the variable X and
click Append.
(4) Compute the media for the even set of
numbers. Examine the ordered series of scores. There are six scores (n = 6). The two middle
values are 6 and 8. To find the half-way between them, add them
together and divide by 2. C = (6 + 8) / 2 = 7
Task B
Throw a coin
15 times. How many heads are you likely
to see? To summarize and describe the result of your experiment, create a
frequency table and a bar chart.
1. Animated Coin
Start a new project. Choose (Animations,
Animated Coin) to start tossing a coin. Click the End button to
terminate the experiment when the number of throws is 15.
The result will appear on the Vector Display.
There are two possible results. Heads are coded as 1. Tails are coded as 0.
Note that the result will not be the same as shown
below due to random chance.
Binary numbers are defined as numbers taking on
only 0 and 1 values. The variable Throw is a binary variable. Count the number
of zeros. Count the number of ones. Choose (Frequencies, Frequencies) and select the
variable Throw to obtain a
frequency table.
Frequency counts can be measured in terms of proportions or
percentages. Next, Choose (Frequencies,
Proportions) and and select the variable Throw.
Create a bar chart
with the number of heads on the horizontal axis and the percentage frequency on
the vertical axis. Choose Graphs.
Select
Bar Graphs under the Graphs Indexed by Attributes category. Abscissa:
arg Prop. Ordinate: Prop, Right click a data value on the Y-axis
to bring up a shortcut menu. Choose Properties. Select Scale tab and
click drop down arrow next to format and select Percentage. Finally,
label the chart and the axes as shown below.
