Making statistics intuitive
Cronbach’s alpha coefficient measures the internal consistency, or reliability, of a set of survey items. Use this statistic to help determine whether a collection of items consistently measures the same characteristic. Cronbach’s alpha quantifies the level of agreement on a standardized 0 to 1 scale. Higher values indicate higher agreement between items. [Read more…] about Cronbach’s Alpha: Definition, Calculations & Example
Cohens d is a standardized effect size for measuring the difference between two group means. Frequently, you’ll use it when you’re comparing a treatment to a control group. It can be a suitable effect size to include with t-test and ANOVA results. The field of psychology frequently uses Cohens d. [Read more…] about Cohens D: Definition, Using & Examples
Statistical inference is the process of using a sample to infer the properties of a population. Statistical procedures use sample data to estimate the characteristics of the whole population from which the sample was drawn.
Scientists typically want to learn about a population. When studying a phenomenon, such as the effects of a new medication or public opinion, understanding the results at a population level is much more valuable than understanding only the comparatively few participants in a study.
Unfortunately, populations are usually too large to measure fully. Consequently, researchers must use a manageable subset of that population to learn about it.
By using procedures that can make statistical inferences, you can estimate the properties and processes of a population. More specifically, sample statistics can estimate population parameters. Learn more about the differences between sample statistics and population parameters.
For example, imagine that you are studying a new medication. As a scientist, you’d like to understand the medicine’s effect in the entire population rather than just a small sample. After all, knowing the effect on a handful of people isn’t very helpful for the larger society!
Consequently, you are interested in making a statistical inference about the medicine’s effect in the population.
Read on to see how to do that! I’ll show you the general process for making a statistical inference and then cover an example using real data.
Related post: Descriptive vs. Inferential Statistics
In its simplest form, the process of making a statistical inference requires you to do the following:
Of course, that’s the simple version. In real-world experiments, you might need to form treatment and control groups, administer treatments, and reduce other sources of variation. In more complex cases, you might need to create a model of a process. There are many details in the process of making a statistical inference! Learn how to incorporate statistical inference into scientific studies.
Statistical inference requires using specialized sampling methods that tend to produce representative samples. If the sample does not look like the larger population you’re studying, you can’t trust any inferences from the sample. Consequently, using an appropriate method to obtain your sample is crucial. The best sampling methods tend to produce samples that look like the target population. Learn more about Sampling Methods and Representative Samples.
After obtaining a representative sample, you’ll need to use a procedure that can make statistical inferences. While you might have a sample that looks similar to the population, it will never be identical to it. Statisticians refer to the differences between a sample and the population as sampling error. Any effect or relationship you see in your sample might actually be sampling error rather than a true finding. Inferential statistics incorporate sampling error into the results. Learn more about Sampling Error.
The following are four standard procedures than can make statistical inferences.
Let’s look at a real flu vaccine study for an example of making a statistical inference. The scientists for this study want to evaluate whether a flu vaccine effectively reduces flu cases in the general population. However, the general population is much too large to include in their study, so they must use a representative sample to make a statistical inference about the vaccine’s effectiveness.
The Monto et al. study* evaluates the 2007-2008 flu season and follows its participants from January to April. Participants are 18-49 years old. They selected ~1100 participants and randomly assigned them to the vaccine and placebo groups. After tracking them for the flu season, they record the number of flu infections in each group, as shown below.
| Treatment | Flu count | Group size | Percent infections |
| Placebo | 35 | 325 | 10.8% |
| Vaccine | 28 | 813 | 3.4% |
| Effect | 7.4% |
From the table above, 10.8% of the unvaccinated got the flu, while only 3.4% of the vaccinated caught it. The apparent effect of the vaccine is 10.8% – 3.4% = 7.4%. While that seems to show a vaccine effect, it might be a fluke due to sampling error. We’re assessing only 1,100 people out of a population of millions. We need to use a hypothesis test and confidence interval (CI) to make a proper statistical inference.
While the details go beyond this introductory post, here are two statistical inferences we can make using a 2-sample proportions test and CI.
For more information about this and other flu vaccine studies, read my post about Flu Vaccine Effectiveness.
In conclusion, by using a representative sample and the proper methodology, we made a statistical inference about vaccine effectiveness in an entire population.
Monto AS, Ohmit SE, Petrie JG, Johnson E, Truscon R, Teich E, Rotthoff J, Boulton M, Victor JC. Comparative efficacy of inactivated and live attenuated influenza vaccines. N Engl J Med. 2009;361(13):1260-7.
The t distribution is a continuous probability distribution that is symmetric and bell-shaped like the normal distribution but with a shorter peak and thicker tails. It was designed to factor in the greater uncertainty associated with small sample sizes.
The t distribution describes the variability of the distances between sample means and the population mean when the population standard deviation is unknown and the data approximately follow the normal distribution. This distribution has only one parameter, the degrees of freedom, based on (but not equal to) the sample size. [Read more…] about T Distribution: Definition & Uses
A representative sample is one where the individuals in the sample reflect the properties of an entire population. Use a representative sample when you want to generalize the results from the sample to a population. By studying a representative sample, you can approximate the properties of the population from which it was drawn. [Read more…] about Representative Sample: Definition, Uses & Methods
The difference between a standard deviation and a standard error can seem murky. Let’s clear that up in this post!
Standard deviation (SD) and standard error (SE) both measure variability. High values of either statistic indicate more dispersion. However, that’s where the similarities end. The standard deviation is not the same as the standard error. [Read more…] about Difference Between Standard Deviation and Standard Error
P values are everywhere in statistics. They’re in all types of hypothesis tests. But how do you calculate a p-value? Unsurprisingly, the precise calculations depend on the test. However, there is a general process that applies to finding a p value.
In this post, you’ll learn how to find the p value. I’ll start by showing you the general process for all hypothesis tests. Then I’ll move on to a step-by-step example showing the calculations for a p value. This post includes a calculator so you can apply what you learn. [Read more…] about How to Find the P value: Process and Calculations
Sampling methods are the processes by which you draw a sample from a population. When performing research, you’re typically interested in the results for an entire population. Unfortunately, they are almost always too large to study fully. Consequently, researchers use samples to draw conclusions about a population—the process of making statistical inferences. [Read more…] about Sampling Methods: Different Types in Research
The beta distribution is a continuous probability distribution that models random variables with values falling inside a finite interval. Use it to model subject areas with both an upper and lower bound for possible values. Analysts commonly use it to model the time to complete a task, the distribution of order statistics, and the prior distribution for binomial proportions in Bayesian analysis. [Read more…] about Beta Distribution: Uses, Parameters & Examples
The geometric distribution is a discrete probability distribution that calculates the probability of the first success occurring during a specific trial. In other words, during a series of attempts, what is the probability of success first occurring during each attempt? Use this distribution when you need to understand how many attempts are necessary to produce the first successful outcome. [Read more…] about Geometric Distribution: Uses, Calculator & Formula
Power in statistics is the probability that a hypothesis test can detect an effect in a sample when it exists in the population. It is the sensitivity of a hypothesis test. When an effect exists in the population, how likely is the test to detect it in your sample? [Read more…] about What is Power in Statistics?
A conditional distribution is a distribution of values for one variable that exists when you specify the values of other variables. This type of distribution allows you to assess the dispersal of your variable of interest under specific conditions, hence the name. [Read more…] about Conditional Distribution: Definition & Finding
A marginal distribution is a distribution of values for one variable that ignores a more extensive set of related variables in a dataset.
That definition sounds a bit convoluted, but the concept is simple. The idea is that when you have a larger set of related variables that you collected for a study, you might want to focus on one of them to answer a specific question. [Read more…] about Marginal Distribution: Definition & Finding
Content validity is the degree to which a test or assessment instrument evaluates all aspects of the topic, construct, or behavior that it is designed to measure. Do the items fully cover the subject? High content validity indicates that the test fully covers the topic for the target audience. Lower results suggest that the test does not contain relevant facets of the subject matter. [Read more…] about Content Validity: Definition, Examples & Measuring
Parameters are numbers that describe the properties of entire populations. Statistics are numbers that describe the properties of samples. [Read more…] about Parameter vs Statistic: Examples & Differences
A spurious correlation occurs when two variables are correlated but don’t have a causal relationship. In other words, it appears like values of one variable cause changes in the other variable, but that’s not actually happening. [Read more…] about Spurious Correlation: Definition, Examples & Detecting
A contingency table displays frequencies for combinations of two categorical variables. Analysts also refer to contingency tables as crosstabulation and two-way tables. [Read more…] about Contingency Table: Definition, Examples & Interpreting
In mathematics and statistics, permutations vs combinations are two different ways to take a set of items or options and create subsets. For example, if you have ten people, how many subsets of three can you make? While permutation and combination seem like synonyms in everyday language, they have distinct definitions mathematically.
Let’s understand this difference between permutation vs combination in greater detail. And then you’ll learn how to calculate the total number of each. [Read more…] about Permutation vs Combination: Differences & Examples
Cumulative frequency is the running total of frequencies in a table. Use cumulative frequencies to answer questions about how often a characteristic occurs above or below a particular value. It is also known as a cumulative frequency distribution.
For example, how many students are in the 4th grade or lower at a school?
Cumulative frequency builds on the concepts of frequency and frequency distribution.
In this post, learn how to find and construct cumulative frequency distributions for both discrete and continuous data. I’ll also show you how to create less than and greater than versions of these tables.
Finding a cumulative frequency distribution makes the most sense when your data have a natural order. The natural ordering allows the cumulative running total to be meaningful. With a minor change, the process works with both discrete and continuous data. Learn more about the differences between Discrete vs. Continuous Data.
For example, the grades in a school, months of a year, or age in years are discrete values with a logical order. Alternatively, when you have continuous data, you can create ranges of values known as classes. In this case, frequencies are counts of how often continuous data fall within each class.
Calculate cumulative frequency by starting at the top of a frequency table and working your way down. Take each row’s frequency and add all preceding rows. By summing the current and previous rows, you calculate the running total.
Let’s use this method to find cumulative frequency for discrete and continuous data.
The example below shows you how to construct a cumulative frequency distribution for a discrete dataset of school grades (1 – 6). Notice how each row takes the previous cumulative frequency and then adds the frequency for that row to calculate the running total.
For example, if we look at the 3rd grade row of the table, we’ll see that the cumulative frequency is 58. This result tells us that 58 students are in the third grade and lower.
In this table, the cumulative frequency for the highest value equals the total number of observations in the dataset because all values are less than or equal to it. 6th grade is the highest value, and 88 students are less than or equal to it. Hence, we know there are 88 students in this dataset.
When you have continuous data, you might not have any repeating values.
For example, no values repeat in the portion of the height data below. Consequently, you’d have a series of values, each having a frequency of one. These are actual data from a study I conducted involving preteen girls. The full dataset has 88 values. You can download the Excel file with the data and table: HeightFrequencyTable.
However, you can obtain meaningful information by grouping the values into ranges and finding the frequency for each class, as shown below.
Then, to create the cumulative frequency table, sum each row with all preceding rows just as we did for the discrete data example.
For example, by looking at the row for 1.46 – 1.51m, we know that 49 preteen girls (just over half the sample of 88) have heights that are less than or equal to 1.51m.
Both the preceding examples use the “less than” form of the table. When you look at those cumulative frequency tables, the value indicates the total number of observations that are less than or equal to a specific value. For example, 70 students are in 4th grade or lower.
However, what should you do when you need to understand frequencies that are greater than or equal to a particular value? Simply switch the order of values in the table to list them from highest to lowest. This process constructs a greater than cumulative frequency distribution.
In the example below, I’ll recreate the grade level table, but instead of listing the grades 1 → 6, I’ll switch it to 6 → 1. From that point, I’ll use the same method of summing the current row with all previous rows.
In this greater than distribution, the cumulative frequencies indicate the number of observations greater than or equal to a particular value. For example, 30 students are in 4th grade or higher.
In this table, the cumulative frequency for the lowest value equals the total number of observations because all observations are greater than or equal to it. 1st grade is the lowest value with 88 students greater than or equal to it.
The decision to use a less than or greater than cumulative frequency table depends on which form is most helpful for your subject area.
You can also show cumulative frequency on graphs. In the bar chart below, I added the orange cumulative line. By displaying it in a chart, it’s easy to find where most observations occur. Learn more about Bar Charts: Using, Examples and Interpreting.
In the graph, first and second graders comprise nearly half the school. As the grade levels progress from low to high, the orange line rises to the total number of students, 88.
Relative frequencies are a related concept. Click the link to learn about similarities and differences!
The chi-square goodness of fit test evaluates whether proportions of categorical or discrete outcomes in a sample follow a population distribution with hypothesized proportions. In other words, when you draw a random sample, do the observed proportions follow the values that theory suggests. [Read more…] about Chi-Square Goodness of Fit Test: Uses & Examples