How To Change Stem Width In Spss

ii: Stem-&-Leaf Plots, Frequency Tables, and Histograms

Stem-and-Leaf Plots
Frequency Tables
� Raw Data � Uniform Class Intervals � Nonuniform Class Intervals
Histograms

Stem-and-Foliage Plots

The stalk-and-leaf plot is an excellent way to outset an analysis. To construct a stalk-and-leaf plot:

(A) Draw a stem-like axis that covers the range of potential values.
(B) Circular the data to two or three significant digits.
(C) Separate each data-signal into a stalk component and leaf component. The stem component consists of all but the rightmost digit; the leafage component consists of the rightmost digit.
(D) Place each leaf value next to its associated stem value, one leaf on top of the other.

To illustrate stem-and-leaf plots, permit us consider a data gear up with the following numerical values:

21, 42, 5, 11, xxx, 50, 28, 27, 24, 52

To start, describe a stem-like axis that extends from the information fix's minimum to its maximum:

|5| |4| |3| |2| |ane| |0| (x 10)

An axis multiplier (ten 10) is included to allow the viewer to decipher the value of each data point.

The rightmost digit of each data point (the "leaf") is then plotted confronting the stem-like axis. For example, a value 21 is plotted equally:

|five| |4| |3| |2|1 |1| |0| (x ten)

The remaining data points are plotted:

|5|02 |4|2 |3|0 |2|1874 |1|1 |0|5 (x 10)

Data are now sorted in judge rank order, and the shape, location and spread of the distribution are evident. I'grand going to flip the stalk-and-leafage to a horizontal orientation to amend display these features.

The location of the data can be summarized by its center. For instance, the center of the above stalk-and-leaf plot is located between 20 and thirty.

four 7 eight two 5 1 ane 0 2 0 ------------ 0 ane 2 iii four five ------------ ^ Center

The spread of the distribution is seen equally the dispersion of values around the distribution's eye.

4 7 eight 2 five 1 1 0 2 0 ------------ 0 1 2 3 4 5 ------------ <----|----> Spread

The shape of the distribution can be seen equally a "skyline silhouette" of the data.

X X X X X X X X X X ------------ 0 ane 2 three iv 5 ------------

Notice the "skyscraper" in the middle of the distribution. This meridian represents the distribution's "mode." The mode of this particular information set in the interval 20 to 30. Also discover that the data demonstrate pretty-good symmetry around the way. (Nothing is perfect in statistics, especially when the sample is minor.)

With a little practice, a distribution's shape, location, and spread tin can be visualized through the stalk-and-leaf plot.

Second Illustrative Instance of a Stem-and-Leaf Plot: The next illustrative case shows how a stalk-and-leafage plot can be modified to accommodate data that might not immediately lend itself to this blazon of plot. Consider this new data gear up:

one.47, 2.06, 2.36, iii.43, three.74, 3.78, three.94

These data accept 3 significant digits and a decimal point. In such instances, we commencement round the information to 2 significant digits. The information set rounded to two significant digits is: {1.5, 2.i, 2.4, 3.4, 3.7, 3.8, 3.9}. When plotting these points, decimal points are ignored. Using stalk values of i, 2, and iii, we plot:

|one|v |2|xiv |3|4789 (x 1)

Realizing that this plot is somewhat squashed, we could spread it out by splitting the stem using double values with the starting time value reserved for leaf-values betwixt 0 and iv and the second stem-value reserved for leafage-values between 5 and ix. Hither'southward the aforementioned information with double stalk-values:

|1| |ane|five |2|xiv |2| |iii|4 |iii|789 (x 1)

This shows that there is more than than one correct fashion to plot a stem-and-leaf diagram for a given information set.

SPSS: To create a stem-and-leaf plot in SPSS, click on Statistics | Summarize | Explore and select the variable you want a stem-and-leaf plot of in the "Dependent List" dialogue box.

Frequency Tables

Raw Data

It is often useful to consider data in the grade of a frequency table. Frequency tables may include three different types of frequencies. These are:

Frequency counts: The number of times a value occurs in a data set.
Relative frequencies: Frequencies expressed every bit percentages of the total.
Cumulative frequencies: Relative frequencies up to and including the current rank-ordered value.

An example of a table of AGE frequencies is:

Table i. Age (in years), respiratory health survey respondents.

To create a frequency table:

(A) List all potential values in ascending lodge

(B) Tally frequency counts (f_i ) with tick marks or some other accounting mechanism. Listing these frequencies in the Freq column of the table.

(D) Calculate relative frequencies (percentages) for each value (p_i = f_i / due north).

(E) Calculate cumulative frequencies by calculation the cumulative frequency from the prior level to the relative frequency of the current level (c_i = p_i + c_i _-1).

SPSS: To create a frequency table in SPSS, click on Statistics | Summarize | Frequencies and select the variable you want a frequency table of in the "Variable(south):" dialogue box.

As an additional example, permit us reconsider the information fix {21, 42, 5, 11, 30, 50, 28, 27, 24, 52}. A frequency tabular array for this data prepare is:

Value Talley Freq. RelFreq CumFreq ------ ------ ----- -------- ------- 5 / 1 10% 10% eleven / one 10% 20 21 / ane 10% 30% 24 / 1 10% 40% 27 / 1 10% 50% 28 / 1 ten% 60% 30 / 1 ten% 70% 42 / ane 10% 80% 50 / 1 x% xc% 52 / 1 10% 100% ------------------------------------------ TOTAL x 100% --

Because of the modest size of the sample, this frequency tabular array is not particularly useful. For information technology to get more than useful, information must exist grouped into form intervals.

Uniform Class Intervals

It is often hard to acquire much past looking at a frequency list of values when the dataset is small and then that each value in the data set appears simply in one case or twice. To address this problem, we condense the data into class interval groupings.

There are no hard-and-fast rules for determining appropriate class intervals. Nevertheless, here are some rules-of-pollex by which to begin:

(A) Decide on an advisable number of class-interval groupings: The optimum number of class groupings will depend on the range of values and the size of the data ready. In general, large data sets tin support a large number of class groupings and pocket-size information sets can back up fewer. Deciding on a suitable number of class-intervals may require some trial and fault. To start, endeavor course-intervals that are of equal and convenient length (e.1000., 10-year age intervals) or have substantive meaning (due east.k., hypotensive / normotensive / borderline hypertensive / hypertensive). Normally, 4 to 12 grade-intervals is unremarkably sufficient.

(B) Make up one's mind the class interval width. This can be determined with the formula:

Class-interval width = (maximum value - minimum value) / (no. of desired class groupings)

For example, to create eight class groupings for a data fix with a maximum of 19 and minimum of 3, the class interval width = (nineteen - 3) / eight = two.

(C) Fix endpoint conventions. If an observation falls on the boundary betwixt 2 form intervals, nosotros need know in which class interval it will be counted. The two choices are to: (a) include the left boundary and exclude the right boundary or (b) include the right boundary and exclude the left boundary. When faced with this choice, we will utilise the pick "a". For example, when consideration a two unit interval of 2 to 4, we volition exclude the right boundary of 4, so that the interval is between 2 (inclusive) up to 4 (exclusive).

(D) Tabulate the data: One time boundaries are established, the information are counted in the usual way.

A frequency table for the modest data set {21, 42, 5, 11, 30, fifty, 28, 27, 24, 52} with xv-year age form-interval grouping tin can now e shown:

Range Tally Freq. RelFreq CumFreq ------ ------ ----- -------- ------- 0-14 // 2 20% xx% 15-29 //// 4 40% 60% 30-44 // ii twenty% 80% 45-54 // 2 twenty% 100% ------------------------------------------ Full 10 100% --

SPSS: To grouping information in SPSS, click on Transform | Recode | Into Different Variable. This will allow you to fix ranges to serve equally class intervals. Later on recoding the data to these new form intervals (ranges), a Statistics | Summarize | Frequencies command tin be directed against the newly recoded variable.

Nonuniform Class Intervals

At times we might desire to use nonuniform class-intervals when describing frequencies. For example, we may desire to await at age distribution of children with ages grouped as preschool (two-iv years), elementary school (5-11-years), middle-schoolhouse (12-13-years), and highschool (fourteen-19-years). The data from Table 1 can now exist displayed as follows:

Histograms

The investigator may choose to graphically review frequencies in the grade of a histogram. Histograms are bar charts that display frequencies or relative frequencies in the form of face-to-face (touching) bars. A histogram of the historic period distribution data (before it was condensed into class intervals) is shown in the figure to the correct:

SPSS: To create a histogram, click on Graphs | Histogram. To create a frequency polygon with SPSS, click on Graphs | Line | Ascertain, then select the variable yous want to graph in the Category Axis dialogue box. (The line will represent the number of cases, by default.)

Notation

n - sample size
f_i = frequency, value or interval i
p_i = relative frequency, value or interval i
c_i = cumulative frequency, value or interval i

Vocabulary

Cumulative frequency: the accumulation of relative frequencies up to and including the rank-ordered value or class.
Frequency: the number of times a particular detail occurs.
Histogram: a bar graph of frequencies or relative frequencies in which bars bear upon.
Relative frequency: frequencies expressed as a percentage of the total.
Stem-and-leaf plot: a histogram-like display of raw values in a information set.