PCAT Exam: Quantitative Reasoning - A Simple Guide To Probability and Statistics

In general, probability and statistics deal with relationships between quantities, which may not be exact. For example, a graduating student’s salary tends to be higher when that student has a higher GPA, but this correlation need not always be the case. People who exercise regularly tend to live longer than those who do not, but once again, this is not always the case.

Probability is the way in which one measures these types of tendencies.

Statistics, which is closely related, can be broken into two main parts: descriptive statistics and inferential statistics.

Descriptive statistics is the process of analyzing the tendencies of a given population, and inferential statistics is the process of using such an analysis of a sample group to make inferences about the population as a whole. Descriptive statistics includes such things as computing the average score on a test administered to a class. Inferential statistics includes such things as polling a few hundred people in a city to find out their salaries, and then trying to use this information to determine the likely salaries of other people in the city.

Measures of Central Tendency
Given a set X of data points , one would like to have some way to describe the general tendencies of these data.

One of the first questions that can be asked is where the “center” of the data lies.

There are a few different ways to measure this. Each describes a slightly different notion of the “center.”
The first measure the central tendency of a data set is the arithmetic mean, or the average, of the data. The definition of the mean is the following: first add up all the data points, and then divide by the total number of data points in our data set.

A convenient way to write this out is to use summation notation.

To be precise, the mean of the data set X is written as

.

As an example, consider a test given to five students. Suppose the scores on the test were 55, 65, 65, 75, 80, 85, 90, 100.

Then, the average test score is the following:

.
The mean is particularly useful for describing the central tendency when the distribution of data is close to normal, which means that the frequency of different outcomes among our data has a single peak and that the data is approximately equally distributed on both sides of that peak.

However, the mean can be somewhat less useful in situations when the data are divided into several different groups that are widely separated, or when there are some outliers, which are data points that are very far from the rest of the data.

For an example of widely separated data groups, consider a group of six people.

Suppose three of these people make $20,000 per year, and three make $100,000 per year.

Then the average income of these six people, in dollars, is.

However, none of the people actually makes anything close to $60,000 per year.

For an example of how outliers can throw off data, consider a group of nine students who take a test.

Suppose four get a score of 90, four get a score of 100, and one gets a score of 0. Then the average is .

However, the average of the top five students is 95. So, this outlier strongly affects the mean.

Another measurement of central tendency that can be defined is the median.

Given a data set X consisting of data points , the median is defined as the value of the data point in the center, in the sense that half the data lies before it and half lies after it.

Therefore, if n is odd, the median is defined as .

However, if n is even, the median is defined as , which is the mean of the two data points closest to the middle of the data points.

Consider a group of 5 people whose ages are 19, 21, 22, 22, and 25. The median age of this group is 22.

Although outliers can have a substantial effect upon the mean of a data set, they usually do not change the median very much. For example, consider the following list of numbers: 2, 4, 5, 6, 6, 7. This list has a mean of 5 and a median of 5.5

. Now suppose the last number in the list is changed to 99, so that the list is now 2, 4, 5, 6, 6, 99.

This list has an average of , but the median is still 5.5.

One final measure of central tendency that is defined for X is the mode. The mode is defined as the data point that appears most frequently in the data set; if two or more data points are tied for the most frequent appearance, each is defined as a mode.

This means a data set might have multiple modes.

For instance, in the following list of numbers, 55, 60, 65, 65, 70, 80, 85, 85, 90, 95, there are two modes: 65 and 85.

Variation
The next problem in statistics is how to measure the degree to which the data are spread out.

Given a data set X with data points , the variance of X is defined as (recall that indicates the mean).

In other words, the variance of X is the mean of the squares of the differences between each data point and the mean of X.

Given a data set X with data points , define the standard deviation of X as .

In other words, the standard deviation is the positive square root of the variance.

The symbol for the standard deviation is the Greek lowercase letter sigma.

The variance and the standard deviation are both measures of how much the data points are spread out.

A low variance or standard deviation means that the data are clumped up closely, and a large variance or standard deviation generally means that the data are either very spread out or that there are a substantial number of outliers.

Consider an example of computing the variance and the standard deviation.

Suppose a problem asks for the standard deviation for the data set {2, 3, 3, 4}.

Begin by computing the mean, which is .

Now, continue by computing the variance, which is, in summation notation:








Therefore, the variance is .

To find the standard deviation, take the square root:

One last measurement of the variation of a data set is the range.

The range is the difference between the largest and the smallest values in the set.

Graphical Forms of Data
In some cases, it can be very useful to visualize statistical data by graphical means. Statistical data are typically visualized by making a graph where the horizontal axis represents the possible values that the data can take, and the vertical axis represents the frequency with which the data takes that value. These graphs are called frequency plots.

Suppose there is a situation in which the data takes the values 1, 2, 3, 4, and 5.

Suppose there are 4 instances of 1, 5 instances of 2, 7 instances of 3, 5 instances of 4, and 6 instances of 5.

A frequency plot for this situation might look like the following:


From the frequency plot, it is possible to immediately read off the modes of the data: the modes are the values that have the tallest columns. In this case, the mode is 3. Given a frequency plot, it is possible to easily read off the number of data points that take a given value by checking the height of the column for that value. For this reason, frequency plots are convenient ways of quickly displaying the data.

To determine the median, note that there is a total of 27 data points here, so the median is the thirteenth data point. The thirteenth data point in this case has a value of 3, so the median is 3.
To find the mean, use the formula given above.

In this case, the formula results in the following:

= 3.15

Interpreting Displays of Data
A set of data can be visually displayed in various forms to allow for quick identification of characteristics of the set. Histograms, such as the one shown below, display the number of data points (vertical axis) that fall into given intervals (horizontal axis) across the range of the set. The histogram below displays the heights of black cherry trees in a certain city park. Each rectangle represents the number of trees with heights between a given five-point span. For example, the furthest bar to the right indicates that two trees are between 85 and 90 feet.

Histograms can describe the center, spread, shape, and any unusual characteristics of a data set.


A box plot, also called a box-and-whisker plot, divides the data points into four groups and displays the five-number summary for the set, as well as any outliers.

The five-number summary consists of:
- The lower extreme: the lowest value that is not an outlier
- The higher extreme: the highest value that is not an outlier
- The median of the set: also referred to as the second quartile or
- The first quartile or : the median of values below
- The third quartile or : the median of values above
To construct a box (or box-and-whisker) plot, the five-number summary for the data set is calculated as follows:

the second quartile () is the median of the set.

The first quartile () is the median of the values below .

The third quartile () is the median of the values above .

The upper extreme is the highest value in the data set if it is not an outlier (greater than 1.5 times the interquartile range:- ).

The lower extreme is the least value in the data set if it is not an outlier (more than 1.5 times lower than the interquartile range).

To construct the box-and-whisker plot, each value is plotted on a number line, along with any outliers.

The box consists of and as its top and bottom and as the dividing line inside the box.

The whiskers extend from the lower extreme to and from to the upper extreme.


Suppose the box plot displays IQ scores for 12th grade students at a given school.

The five-number summary of the data consists of: lower extreme (67); upper extreme (127); or median (100); (91); (108); and outliers (135 and 140).

Although all data points are not known from the plot, the points are divided into four quartiles each, including 25% of the data points.

Therefore, 25% of students scored between 67 and 91, 25% scored between 91 and 100, 25% scored between 100 and 108, and 25% scored between 108 and 127. These percentages include the normal values for the set and exclude the outliers.

This information is useful when comparing a given score with the rest of the scores in the set.

A scatter plot is a mathematical diagram that visually displays the relationship or connection between two variables.

The independent variable is placed on the -axis, or horizontal axis, and the dependent variable is placed on the y-axis, or vertical axis.

When visually examining the points on the graph, if the points model a linear relationship, or a line of best-fit can be drawn through the points with the points relatively close on either side, then a correlation exists.

If the line of best-fit has a positive slope (rises from left to right), then the variables have a positive correlation. If the line of best-fit has a negative slope (falls from left to right), then the variables have a negative correlation.

If a line of best-fit cannot be drawn, then no correlation exists. A positive or negative correlation can be categorized as strong or weak, depending on how closely the points are graphed around the line of best-fit.



 

Probability
Given a set of possible outcomes X, a probability distribution on X is defined to be a function that assigns a probability to each possible outcome.

If the set X consists of , and the probability distribution is the function p, then the following must be true of p:
 

, for any

The first rule says that p must always take a value between zero and one.

The second rule says that the total probability for all the possible outcomes in X is one.
If is a constant function, then this is called a uniform probability distribution, and , where n is the total number of possible outcomes.

A good example of this is a fair 6-sided die, in which the possible outcomes are 1, 2, 3, 4, 5, and 6, and the probability of each of these 6 outcomes is .

To determine the probability of an outcome occurring from a range A of possible outcomes, write this probability as .

To compute this, add up the probabilities for each outcome in A.

To use the example of a fair 6-sided die, consider the problem of finding the probability of getting a 2 or a lower number when the die is rolled.

The possible rolls are 1, 2, 3, 4, 5, and 6. So to get a 2 or lower, one must roll a 1 or a 2.

Each probability is ; add the probablities together to get .

Below are a few types of probability distributions that are standard and have standard names.

The binomial distribution: This distribution describes the probability of getting k successes in n trials, where each trial can either succeed or fail.

Suppose the probability of a single trial’s being a success is p. Then, the probability of getting k successes in n trials is:

A Poisson distribution: This describes the probability of getting k events in a fixed interval of time.

Suppose the average number of events during this time interval is λ.

Then, the probability of getting k events during this time interval is given by:



Conditional Probabilities
In some cases, it may be known that the outcome lies within the subset of possibilities B, and a problem asks for the probability that the outcome will be inside another subset of possibilities A.

This kind of problem concerns a conditional probability.

A conditional probability is written as , which is read as “the probability of A given B.”

The formula for this quantity in general is .
In the case of uniform probability distributions, it is possible to simplify this formula somewhat. It is standard notation to write to indicate the total number of outcomes in A.

Then, if one is dealing with a uniform probability distribution, it is possible to write (remember, means “A intersect B,” and is the set of all outcomes that lie in both A and B).

So, in this case there is no need to know how many total outcomes there might be; all that one needs to know is the total in B and the total in .
To understand why this works, suppose that a set of outcomes X is , so that , and the probability distribution on X is a uniform probability distribution p.

Then for each .

Then, from the definition of a uniform probability distribution, .

Similar formulas are obtained for and .

Substituting this formula into the formula for conditional probabilities results in , because the n’s cancel out.

As an application of this principle, suppose a fair die is rolled. It is known that the roll’s value lies between 1 and 4, inclusive, but the number of sides that the die has is not known (not all dice are 6-sided).

 

Suppose the problem asks for the probability that the roll was higher than 2.

In this case, the total number of sides of the die is unimportant, and, because this is a fair die, the probability distribution is uniform across all possibilities.

Therefore, it is possible to apply the formula .

In this particular problem, B is {1, 2, 3, 4} and is {3, 4}.

Therefore .

Conditional probability is very important in probability because often the likelihood of one outcome can change substantially after some additional information about the outcome is given. For example, if someone wants to know the probability that a student will pass a test, they could look at the statistics about how many students usually pass that test. However, if they also know that the student in question has spent a lot of time studying, then they know it is much more likely the student will pass (although nothing is guaranteed).

In many cases, changing the order of the conditional probabilities greatly affects the outcome. Consider determining the probability that a person with heart trouble is a person who has exercised regularly, versus the probability of a person who does regular exercise coming down with heart trouble. As another example, given a person who has received a military medal, it is certain that the person served in the military. However, given that a person served in the military, the probability of their receiving a military medal may not be very high.

In some special cases, however, the order in which the conditional probabilities are taken does not change the final probability.

 

Consider the situation in which the probability of A does not change when B is given, that is, the situation where .

In this case, A and B are said to be independent.

The fact means that if A and B are independent, .

From the equation , one can perform this computation in reverse to also show that , and .

Therefore, when A is independent of B, B is also independent of A.

A situation in which one would expect the outcomes to be independent is in rolling a pair of dice.

Suppose that one rolls 2 dice: a white die and a black die. Then, one would expect that the number that is obtained from the white die should not depend upon the number that is obtained from the black die, or vice versa. The same principle applies to a situation in which one rolls a single die repeatedly. A similar situation applies to flipping a coin repeatedly: whether the next flip is heads or tails does not depend upon the results of previous flips.
Of course, this can be counterintuitive, because in cases as rolling a die or flipping a coin, there is occasionally a series of surprising results, for example, a person might get 5 tails in a row while flipping a coin. But if the probability distribution is already known, then this occurrence is simply random. In fact, if a person keeps flipping a coin or rolling a die for a long while, it is very likely that some unlikely series of outcomes will sometimes occur.
 

A similar mistake in reasoning about probabilities is the idea that a low-probability outcome is necessarily surprising. Of course, getting a low-probability outcome in a single try would be surprising, but in reality, there are so many things going on at all times that there are almost certain to be a few low-probability occurrences.

A good illustration of this phenomenon can be seen in lotteries. The odds of a given person’s winning the lottery are low. On the other hand, the odds that some person will win the lottery each week are reasonably high, and people are not surprised when they find out that somebody has won. For a similar reason, one should, in general, not be too surprised by the occurrence of some seemingly-unlikely events.

Statistical Concepts
Statistics involves making decisions and predictions about larger sets of data based on smaller data sets. The information from a small subset can help predict what happens in the entire set. The smaller data set is called a sample and the larger data set for which the decision is being made is called a population.

The three most common types of data-gathering techniques are sample surveys, experiments, and observational studies.

Sample surveys involve collecting data from a random sample of people from a desired population. The measurement of the variable is only performed on this set of people. To have accurate data, the sampling must be unbiased and random. For example, surveying students in an advanced calculus class on how much they enjoy math classes is not a useful sample if the population should be all college students based on the research question. There are many methods to form a random sample, and all adhere to the fact that every sample that could be chosen has a predetermined probability of being chosen. Once the sample is chosen, statistical experiments can then be carried out to investigate real-world problems.

An experiment is the method in which a hypothesis is tested using a trial-and-error process. A cause and the effect of that cause are measured, and the hypothesis is accepted or rejected.

Experiments are usually completed in a controlled environment where the results of a control population are compared to the results of a test population. The groups are selected using a randomization process in which each group has a representative mix of the population being tested. Finally, an observational study is similar to an experiment. However, this design is used when there cannot be a designed control and test population because of circumstances (e.g., lack of funding or unrealistic expectations). Instead, existing control and test populations must be used, so this method has a lack of randomization.

A statistical question is answered by collecting data with variability. Data consists of facts and/or statistics (numbers), and variability refers to a tendency to shift or change. Data is a broad term, inclusive of things like height, favorite color, name, salary, temperature, gas mileage, and language. Questions requiring data as an answer are not necessarily statistical questions. If there is no variability in the data, then the question is not statistical in nature. Consider the following examples: what is Mary’s favorite color? How much money does your mother make? What was the highest temperature last week? How many miles did your car get on its last tank of gas? How much taller than Bob is Ed?

None of the above are statistical questions because each case lacks variability in the data needed to answer the question. The questions on favorite color, salary, and gas mileage each require a single piece of data, whether a fact or statistic. Therefore, variability is absent. Although the temperature question requires multiple pieces of data (the high temperature for each day), a single, distinct number is the answer. The height question requires two pieces of data, Bob’s height and Ed’s height, but no difference in variability exists between those two values. Therefore, this is not a statistical question. Statistical questions typically require calculations with data.

Consider the following statistical questions:
How many miles per gallon of gas does the 2016 Honda Civic get? To answer this question, data must be collected. This data should include miles driven and gallons used. Different cars, different drivers, and different driving conditions will produce different results. Therefore, variability exists in the data. To answer the question, the mean value could be determined.
 

Are American men taller than Chinese men? To answer this question, data must be collected. This data should include the heights of American men and the heights of Chinese men. All American men are not the same height and all Chinese men are not the same height. Some American men are taller than some Chinese men and some Chinese men are taller than some American men. Therefore, variability exists in the data. To answer the question, the median values for each group could be determined and compared.

Interpreting Statistical Information
To make decisions concerning populations, data must be collected from a sample. The sample must be large enough to be able to make conclusions. A common way to collect data is via surveys and polls. Every survey and poll must be designed so that there is no bias. An example of a biased survey is one with loaded questions, which are either intentionally worded or ordered to obtain a desired response. Once the data is obtained, conclusions should not be made that are not justified by statistical analysis.

One must make sure the difference between correlation and causation is understood. Correlation implies there is an association between two variables, but it does not imply causation.

Population Mean and Proportion
Both the population mean and proportion can be calculated using data from a sample.

The population mean () is the average value of the parameter for the entire population.

Due to size constraints, finding the exact value of is impossible, so the mean of the sample population is used as an estimate instead.

The larger the sample size, the closer the sample mean gets to the population mean.

An alternative to finding is to find the proportion of the population, which is the part of the population with the given characteristic.

The proportion can be expressed as a decimal, fraction, or percentage, and can be given as a single value or a range of values.

Because the population mean and proportion are both estimates, there’s a margin of error, which is the difference between the actual value and the expected value.

T-Tests
A randomized experiment is used to compare two treatments by using statistics involving a t-test, which tests whether two data sets are significantly different from one another.

To use a t-test, the test statistic must follow a normal distribution.

The first step of the test involves calculating the t value, which is given as where and  are the averages of the two samples.

Also, where and are the standard deviations of each sample and and are their respective sample sizes.

The degrees of freedom for two samples are calculated as rounded to the lowest whole number.

Also, a significance level must be chosen; the typical value is

Once everything is compiled, the decision is made to use either a one-tailed test or a two-tailed test.

If there’s an assumed difference between the two treatments, a one-tailed test is used. If no difference is assumed, a two-tailed test is used.

Analyzing Test Results
Once the type of test is determined, the t-value, significance level, and degrees of freedom are applied to the published table showing the t distribution. The row is associated with degrees of freedom and each column corresponds to the probability.

The t-value can be exactly equal to one entry or lie between two entries in a row.

For example, consider a t-value of 1.7 with degrees of freedom equal to 30. This test statistic falls between the p values of 0.05 and 0.025. For a one-tailed test, the corresponding p value lies between 0.05 and 0.025.

For a two-tailed test, the p values need to be doubled, so the corresponding p value falls between 0.1 and 0.05.

Once the probability is known, this range is compared to If , the hypothesis is rejected. If , the hypothesis isn’t rejected.

In a two-tailed test, this scenario means the hypothesis is accepted that there’s no difference in the two treatments.

In a one-tailed test, the hypothesis is accepted, indicating that there’s a difference in the two treatments.

Sample Statistics
A point estimate is a single value used to approximate a population parameter. The sample proportion is the best point estimate of the population proportion. It is used because it is an unbiased estimator, meaning that it is a statistic that targets the value of the population parameter by assuming the mean of the sampling distribution is equal to the mean of the population distribution. Other unbiased estimators include the mean and variance. Biased estimators do not target the value of the population parameter, and such values include median, range, and standard deviation. A confidence interval consists of a range of values that is utilized to approximate the true value of a population parameter. The confidence level is the probability that the confidence interval does contain the population parameter, assuming the estimation process is repeated many times.

Population Inferences Using Distributions
Samples are used to make inferences about a population. The sampling distribution of a sample mean is a distribution of all sample means for a fixed sample size, n, which is part of a population. Depending on different criteria, either a binomial, normal, or geometric distribution can be used to determine probabilities. A normal distribution uses a continuous random variable, and is bell-shaped and symmetric. A binomial distribution uses a discrete random variable, has a finite number of trials, and only has two possible outcomes: a success and a failure. A geometric distribution is very similar to a binomial distribution; however, the number of trials does not have to be finite.

Linear Regression
Regression lines are a way to calculate a relationship between the independent variable and the dependent variable. A straight line means that there’s a linear trend in the data. Technology can be used to find the equation of this line (e.g., a graphing calculator or Microsoft Excel®). In either case, all of the data points are entered, and a line is “fit” that best represents the shape of the data. Other functions used to model data sets include quadratic and exponential models.

Regression lines can be used to estimate data points not already given. For example, if an equation of a line is found that fit the temperature and beach visitor data set, its input is the average daily temperature and its output is the projected number of visitors. Thus, the number of beach visitors on a 100-degree day can be estimated. The output is a data point on the regression line, and the number of daily visitors is expected to be greater than on a 96-degree day because the regression line has a positive slope.

Plotting and Analyzing Residuals
Once the function is found that fits the data, its accuracy can be calculated. Therefore, how well the line fits the data can be determined.

The difference between the actual dependent variable from the data set and the estimated value located on the regression line is known as a residual. Therefore, the residual is known as the predicted value minus the actual value

A residual is calculated for each data point and can be plotted on the scatterplot.

If all the residuals appear to be approximately the same distance from the regression line, the line is a good fit. If the residuals seem to differ greatly across the board, the line isn’t a good fit.

Interpreting the Regression Line
The formula for a regression line is where is the slope and is the y-intercept.

Both the slope and y-intercept are found in the Method of Least Squares, which is the process of finding the equation of the line through minimizing residuals.

The slope represents the rate of change in y as gets larger.

Therefore, because y is the dependent variable, the slope actually provides the predicted values given the independent variable. The y-intercept is the predicted value for when the independent variable equals zero. In the temperature example, the y-intercept is the expected number of beach visitors for a very cold average daily temperature of zero degrees.

The correlation coefficient (r) measures the association between two variables. Its value is between -1 and 1, where -1 represents a perfect negative linear relationship, 0 represents no relationship, and 1 represents a perfect positive linear relationship.

A negative linear relationship means that as -values increase, y values decrease.

A positive linear relationship means that as -values increase, y-values increase.

The formula for computing the correlation coefficient is:



is the number of data points

Both Microsoft Excel® and a graphing calculator can evaluate this easily once the data points are entered.

A correlation greater than 0.8 or less than -0.8 is classified as “strong,” while a correlation between -0.5 and 0.5 is classified as “weak.”

Here is an example of a data set and its regression line:


Regression models are highly used for forecasting, and linear regression techniques are the simplest models. If the nonlinear data follows the shape of exponential, logarithmic, or power functions, those types of functions can be used to more accurately model the data rather than lines.

Here is an example of both an exponential regression and a logarithmic regression model:


A set of data can be described in terms of its center, spread, shape and any unusual features. The center of a data set can be measured by its mean, median, or mode. The spread of a data set refers to how far the data points are from the center (mean or median). The spread can be measured by the range or the quartiles and interquartile range. A data set with data points clustered around the center will have a small spread. A data set covering a wide range will have a large spread.
When a data set is displayed as a histogram or frequency distribution plot, the shape indicates if a sample is normally distributed, symmetrical, or has measures of skewness or kurtosis. When graphed, a data set with a normal distribution will resemble a bell curve.


If the data set is symmetrical, each half of the graph when divided at the center is a mirror image of the other.

If the graph has fewer data points to the right, the data is skewed right. If it has fewer data points to the left, the data is skewed left.


Kurtosis is a measure of whether the data is heavy-tailed with a high number of outliers, or light-tailed with a low number of outliers.
A description of a data set should include any unusual features such as gaps or outliers. A gap is a span within the range of the data set containing no data points. An outlier is a data point with a value either extremely large or extremely small when compared to the other values in the set.

Correlation Versus Causation
Correlation and causation have two different meanings. If two values are correlated, there is an association between them. However, correlation doesn’t necessarily mean that one variable causes the other. Causation (or “cause and effect”) occurs when one variable causes the other.

Average daily temperature and number of beachgoers are correlated and have causation. If the temperature increases, the change in weather causes more people to go to the beach.

However, alcoholism and smoking are correlated but don’t have causation. The more someone drinks, the more likely they are to smoke, but drinking alcohol doesn’t cause someone to smoke.