Getting Ready for AP STATISTICS: LEVELS OF MEASUREMENT

Data can be separated into categories according to what is being measured and what information can be determined from the data.  Different levels of measurement are appropriate for different arithmetic and statistical operations. For example, it makes no sense to find the mean, median, or mode of a list of Social Security numbers.

NOMINAL

The nominal level of measurement is the lowest of the four ways to characterize data because it holds the least potential for instructive analysis. Nominal means "in name only" and nominal data deal with names, categories, or labels -- qualitative groups (see previous post: Getting Ready for AP STATISTICS: VOCABULARY BASICS).

Because data at the nominal level are qualitative, they can't be ordered in a meaningful way, and it makes no sense to calculate things such as the mean or standard deviation.

Examples:
Colors
Yes or no responses to a survey
Favorite class subject
Gender
Makes of cars  

Some number groups can be nominal:  the numbers assigned to marathon runners, for example, are used to identify or name the entrants.

 ORDINAL

The next level is called the ordinal level of measurement. Data at this level CAN be put in some order, but the differences between elements are not meaningful.

Letter grades in your classes are an example of ordinal measurement.  In some schools, a minimum of 94% is required for an A, but another school may count a 90% as an A.  It is not surprising that colleges have difficulty comparing applicants through letter grades alone, or even through class rank.  Both measures tell nothing about the difference between separate elements within a group.

As with the nominal level, data at the ordinal level should not be used in calculations.

Examples:
Movie ratings (G,P-G,PG13,R,NC17)
The Tonight Show’s “TOP 10” of anything
Classroom letter grades
IQ scores

INTERVAL

The third level of measurement is the Interval level.  It deals with data that CAN BE ORDERED, and in which differences between the data DO make sense.  It can be used meaningfully in calculations, but Interval level data DOES NOT HAVE A STARTING POINT.

Temperature for example can be put in increasing or decreasing order and the difference between 20 degrees and 80 degrees makes sense.  Zero degrees, however, does NOT mean a total absence of temperature.  Without a starting point, ratios of element values are meaningless.  An 80 degree day is not four times as hot as a 20 degree day.

Examples:
Temperature
Sea level
A college student's checking account balance (hopefully that's only a joke)

RATIO

The fourth and highest level of measurement is the ratio level where data possess all of the features of the other three levels and also a ZERO VALUE, so it makes sense to compare the ratios of measurements. Phrases such as "four times" and "twice" are meaningful at the ratio level.

Examples:
Monthly precipitation
Any system of measurement (length, weight, distance)




Since organizing data is a constant task in statistics, let's encapsulate characteristics of the four levels of measurement.

Notice that each level of measurement meets the criteria for the previous levels plus one more.

As you get further along in the curriculum, you will be introduced to essential statistical operations.  The level of measurement is an important factor in determining which
computations are viable.  Save the chart below for future reference and check your textbook glossary to reference each of the computations in the left column.

WHAT'S NEXT IN STATISTICS?
Now that we have some background and terminology, we can start developing a statistical survey.  Get a jump start on the classroom by thinking of a study you might want to conduct as a practical application.  Are you in another class (like American Politics during this election year) which would give extra credit for completing the survey?  What is your population?  What sample would be appropriate?  What would you measure?  How would you measure it?  Based on the chart above, what statistical computations would be possible?  Later in the curriculum, we'll investigate the information that can be garnered from the calculations.  For now....have fun!!


                       

PARAMETRIC EQUATIONS: Math Art

Even for those who are not enamored by the intricacies of math, the potential for entering an equation into a grapher and watching an incredible design appear can be appealing.  Parametric equations are the artists of the math world.



Parametric equations appear every now and then on the SATII, in Precalculus classes, and in Calculus as well.  These equations express two variables (usually x and y) each in terms of a third variable called the parameter (often t).

THE OBVIOUS SOLUTIONS
Given the parametric equations....

    x = t^2 + 3
    y = 2t

....a graph can be plotted by calculating values of (x,y) in a three column chart.

    t      x     y
    0     3     0
    1     4     2
    2     7     4
    3    12    6
    4    19    8
    5    28    10

USING OUR ALGEBRA SKILLS
Many parametric equations can also be manipulated algebraically to form a single rectangular equation which may be easier to graph.   

    Solving for t in terms of either x or y in the example above and
    substituting into the alternate equation gives the function...

    y = 2 sqrt (x-3)


A SPECIAL CAUTION WHEN ELIMINATING THE PARAMETER
Caution should be taken to preserve the range of the original equations when eliminating a parameter.  In this example, the range of x is all real numbers greater than or equal to 3.  Luckily, this exclusion on x is also accounted for in the rectangular equation because of the square root.

But consider...

    x = t^2
    y = 3t^2 + 4

In this case, x doesn’t fall below 0 and y doesn’t fall below 4.  So the graph of the parametric equations is truncated below (0,4).

PARAMETERS CAN DO WHAT FUNCTIONS NEVER WILL
Many interesting shapes cannot be graphed as functions because they don’t pass the vertical line test.  Parametric Equations can be used to graph geometric figures like conics.

The Cartesian equation for a circle, x^2 + y^2 = r^2, can easily be converted to parametrics as
    x = r cos t
    y = r sin t

RECOGNIZE CHARACTERISTICS OF PARAMETRICS
Here are some “expectations” for parametric equations.

1.  x = sin t, y = cos t  represents a point moving clockwise around the unit circle.

2.  x = cos t, y = sin t describes the point moving counter-clockwise around the unit circle.
   
3.  x = t, y = t^2 signals a parabola

4.  x = t^2, y = t  in function form graphs as a square root

RELEASE THE ARTIST WITHIN
YouTube has a plethora of examples of exciting, artful parametric equations to program into a graphing calculator.  Here’s one that draws a butterfly......

    x = sin(t) (e^(cos(t)) - 2 cos (4t) - (sin (t/12))^5)
    y = cos(t) (e^(cos(t)) - 2 cos (4t) - (sin (t/12))^5)

Have fun!!

Getting Ready for AP STATISTICS: VOCABULARY BASICS

As you start the study of statistics, you will find topics and equations that you have studied in previous classes.  Mean-median-mode, for example, should quickly bring to mind average-middlemost-most frequent.  Other familiar terms have unique definitions for statistics class.  Meanings aren’t as much “different” as they are more “specific” and carry definite implications. 

What follows is not a complete list, but rather a sampling of the vocabulary with explicit meanings in the course.  There are only 10 here to get you started and a suggestion for continuing to prepare for AP Stats on your own.

Population: The complete set of data elements is termed the population.

Sample: A sample is a portion of a population selected for further analysis.

Parameter: A parameter is a characteristic of the whole population.

Statistic: A statistic is a characteristic of the sample.

                 Notice that the P’s are associated with each other and 
                 the S’s are associated with each other:  
                 Parameter is to Population as Statistic is to Sample

Individual refers to the elements in a set

Variable refers to the characteristics describing individual elements of a set.

Qualitative data are non-numeric or categorical data.

Quantitative data are numeric and can be either Discrete or Continuous.

     Discrete: numeric data that have a finite number of possible values.  When opinion surveys evaluate answers like “Strongly disagree, Disagree, Neither disagree or agree, Agree, Strongly agree, each possible answer is given a discrete numeric value so meaningful statistical computations can be made.  The same information could be gathered through options like “on a scale of 1 to 5...”

Data counts are always discrete: for example, the number of students enrolled in your statistics classes.

     Continuous: numeric data that have infinite possibilities: the set of all counting numbers or the set of all real, rational numbers between 1 and 2.



If you already have your textbook, look at the glossary to find other terms that you've run into before.  Read definitions to identify how these words might be similar and/or different from your past encounters with them. 

Getting Ready for AP STATISTICS: MEAN-MEDIAN-MODE

MEAN - MEDIAN - MODE

For those of us preparing to earn a 5 on the AP Statistics exam next spring, calculating mean - median - mode should be as natural as eating ice cream on a hot summer’s day.  The measures of central tendency were introduced in upper elementary math classes, but it isn’t until now that the various uses have important implications.  

CALCULATING MEASURES OF CENTRAL TENDENCY

MEAN is the same as “average”  or “arithmetic mean.”
     Add all of the elements and divide the sum by the number of elements.

MEDIAN is the middlemost.  Think of the middle of the street...that’s the median. 
One very easy way to find the median in a small group is to first put all numbers in ascending order. Then count from the right and left simultaneously until you get to the middle.

If there are an odd number of elements, there will be one in the middle that can’t be matched from both the right and left......the median.

If there are an even number of elements, the median is the AVERAGE of the two in the middle.

MODE is the most frequently appearing element.

USING CENTRAL TENDENCY TO DESCRIBE DATA

In Statistics, these measures are used to describe the “typical” element and the relative usefulness of each is determined by the level of measurement.  (Watch this site for an upcoming blog, "Getting Ready for AP Statistics: Levels of Measurement" and share examples with other followers.)

NOMINAL (entries are categories or qualitative data)......MODE

ORDINAL (qualitative or quantitative data)....................MEDIAN

INTERVAL (quantitative data)
--- Symmetric.............................................................MEAN
--- Skewed.................................................................MEDIAN


RATIO (quantitative data)
--- Symmetric.............................................................MEAN
--- Skewed.................................................................MEDIAN

A MEAN is useful only for interval or ratio data.  It considers all of the elements in the sample and is influenced by outliers.  The mean is pulled toward an extreme.

A MEDIAN is useful for all quantitative data: ordinal, interval, and ratio levels and is not effected by an extreme outlier.

A MODE is the only useful descriptor for nominal measurements.  It can be applied in all levels of measurement but may not exist or be meaningful in all cases.

 MEAN - MEDIAN - MODE ON A FREQUENCY GRAPH

  

 

USING A CALCULATOR FOR RECORDING MEAN - MEDIAN 

If you're using a TI-84 or similar grapher, check out the "LIST" and "STAT" buttons.  There are options for finding mean and median after listing the elements in a set.  While you're at it, find the owner's manual for the calculator you'll be using in class and look over the directions for inputting a list and calculating various statistical values.

Keep watching this site for more "Previews" of AP Stats, AP Calc, and other classroom courses.  Let's be prepared before school resumes!

GETTING READY FOR THE PSAT

Why think about the PSAT?
There is no greater honor for a high school senior than to be nominated, selected, chosen, recognized as a National Merit Scholar.  The only way to win the honor which represents the very pinnacle of student achievement is to get an outstanding score on the PSAT in  October of Junior year of high school.  The path to that accolade begins long before the actual distinction, however, perhaps as early as fourth or fifth grade.

How does the elementary education impact results on the PSAT?
The simple answer is VOCABULARY.  On the PSAT, two separate vocabulary assessments are included in the two Critical Reading sections.  Missing just 4 of these questions could knock a test taker out of the running by lowering the raw score by 5 points.

At Tutoring Resources we encourage high ability students to begin expanding recognized vocabulary even before entering Middle School.  “Recognized” basically means listening vocabulary; the student doesn’t need to use the words in normal speech or even written work, but hearing the word should create a usable meaning which can be described out of context.   

Vocabulary can be improved regardless of the student’s grade.  Potential National Merit scholars should develop a plan to expand word skills.  A few suggestions include “SAT Word-a-Day” available free online, any one of several commercial programs, avid reading (with a dictionary close by), prefix/suffix knowledge, and listening closely to educational tv programs in search of college-level terminology.  Word-a-Day (or -Week or -Month) projects  can be family activities to benefit siblings of all ages and parents as well.  My grandmother, who learned English after the age of 30, continued to explore vocabulary well into her 90s by recording new words as she ran across them and looking them up later.

When should study for PSAT and National Merit begin in earnest?
The PSAT test serves as a qualifier for National Merit recognition only when taken in October of junior year.  Some students begin preparation in Middle School, others after completing Algebra 2, and a few as they enter eleventh grade.  All 2014 grads should be thinking about this opportunity right now.  Those with the potential to excel should double down on preparation immediately.

How do you know if it is worthwhile to study for the PSAT? 
Take one of the REAL ACT tests out of the study manual of the same name.  Individual section scores over 30 on the English, Math, and Reading portions are good indicators of the necessary foundational knowledge.  Because the structure of the two national tests is noticeably different, the study goal should be to fortify content experience while practicing test taking skills unique to the PSAT/SAT.

A sample SAT score can also serve to motivate a student to strive toward PSAT success.  The scoring protocol parallels the PSAT, and scores of 700 or higher indicate a high probability of sufficient background knowledge to warrant enhanced study toward the October test.

What study goals should be established? 
Students identified at Tutoring Resources as potential National Merit scholars are invited to study for the PSAT.  For Illinois students, to accommodate possible anxiety on test day, we set homework goals that would result in a minimum PSAT score of 220, slightly higher than the 216 cut off established last year .

What if I don’t get picked to continue the National Merit qualification process?
As a certifiable academic, I will never be convinced that learning is not worthwhile.  With that prejudice in mind, I would argue that recognizing a high scholastic aptitude is a reward in itself.  Striving toward the coveted academic award has its own compensation in terms of personal satisfaction and ultimate success on national college entrance exams.  Winning a National Merit Scholarship is the icing on the cake but only a small part of the whole experience.

GOOD LUCK!!!

Getting Ready for AP STATISTICS: COMMON GRAPHS

Statistics is the collection and analysis of data for the purpose of discovering trends, formulating conclusions, and making predictions.  Sometimes "picturing" the data can be useful in explaining relationships.  Several forms of visual display are taught in math classes that come before a formal Statistics class.  

Charts and graphs are used to display specific data, categories, trends, and other visualizations which may make it easier to understand statistical measurements.  The advantages and disadvantages of each display determine the best use for each.

EXAMPLE #1

 

HISTOGRAM 

Individual data points are collected into categories which are represented in adjacent bars.

Advantages:  Visually strong

                      Can compare to normal curve

                      Usually vertical axis is a frequency count of items in a category

Disadvantages:  Cannot read exact values because data is grouped into categories

                           More difficult to compare two data sets

                           Use only with continuous data

EXAMPLE #2

 

 

 

 

 

 

 

BAR CHART

Similar to Histogram but individual data points are represented in each bar, avoiding disadvantage of not being able to distinguish exact values.

EXAMPLE #3

LINE GRAPH

Plots continuous data as points and joins them with a line.  Multiple data sets can be graphed together with separate sets identified in a key.
Advantages:  Can compare multiple
                      continuous data sets
                      Interim data can be inferred
                      from graph line
Disadvantages:  Data must be continuous

EXAMPLE #4

 

FREQUENCY POLYGON 

Made by coloring in the area below a line graph.  From a histogram, construct the polygon by connecting midpoints of each column.
Advantages:  Attractive display
Disadvantages:  Ends may imply zero as
                           data points
                           Use only with continuous
                           data

 

 

EXAMPLE #5

SCATTER PLOT (DOT PLOT)

Displays the relationship between two factors of the experiment.  A line of best fit is used to indicate positive, negative, or lack of correlation.
Advantages:  Trends are easy
                      to see in
                      Shows exact
                      values and entire
                      sample set
                      Shows min, max,
                      and outliers
Disadvantages: Both data sets
                          must be
                          continuous

EXAMPLE #6

 

 

 

 

 

 

 

STEM AND LEAF DISPLAY

Data displayed in rows which can easily be converted to histograms or frequency graphs.  If rows become too long,  left column entries may be repeated.  Strings are most useful if entries are in numeric order.
Advantages:  Precise data list
                      Easily identify min, max, gaps, clusters,
                      and outliers
 Disadvantages:  Central tendencies difficult to visualize

EXAMPLE #7

BOX AND WHISKERS   (BOX PLOT)

Showing the five standard measures: median, upper and lower quartiles, smallest value, and largest value . Multiple boxplots can be drawn side by side to compare more than one data set. 

Advantages
    Shows 5-point summary and outliers
    Easily compares two or more data sets
    Handles extremely large data sets easily

Disadvantages
    Exact values are lost

EXAMPLE #8

Here's a graph that shows up frequently in calculus problems...

 OGIVE (CUMULATIVE LINE GRAPH)

Displays the total at any given time. The relative slopes from point to point will indicate greater or lesser increases; for example, a steeper slope means a greater increase than a more gradual slope.



For some of these visualizations, like the box and whiskers display or a line of best fit, it is necessary to calculate trends or measures of central tendency.  Can you find the mean? the median? the mode? the first quartile?  the standard deviation?

Definitions and calculations for these and other important principles in Stats will be covered in upcoming blogs.  A little review before school resumes can help get the new term off on the right foot.  Keep watching this sight for tips for AP Stats and other courses.