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
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
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
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
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!
A cohort study is a longitudinal experimental design that follows a group of participants who share a defining characteristic. For example, a cohort study can select subjects who have exposure to a risk factor, are in the same profession, population or generation, or experience a particular event, such as a medical procedure. This design determines whether exposure to a risk factor affects an outcome. [Read more…] about Cohort Study: Definition, Benefits & Examples
There are several ways to find the mode depending upon the data type and sample size. In statistics, the mode is the most frequently occurring value in a data set. It is a measure of central tendency. To learn more about the mode, read my post, Measures of Central Tendency. [Read more…] about How to Find the Mode
A bimodal distribution has two peaks. In the context of a continuous probability distribution, modes are peaks in the distribution. The graph below shows a bimodal distribution. [Read more…] about Bimodal Distribution: Definition, Examples & Analysis
Quartiles are three values that split your dataset into quarters. [Read more…] about Quartile: Definition, Finding, and Using
Construct validity relates to the soundness of inferences that you draw from test scores and other measurements. Specifically, it addresses whether a test measures the intended construct. For example, does a test that evaluates self-esteem truly measure that construct or something else? [Read more…] about Construct Validity: Definition and Assessment
Qualitative research aims to understand ideas, experiences, and opinions using non-numeric data, such as text, audio, and visual recordings. The focus is on language, behaviors, and social structures. Qualitative researchers want to present personal experiences and produce narrative stories that use natural language to provide meaningful answers to their research questions. [Read more…] about Qualitative Research: Goals, Methods & Benefits
The definition of a variable changes depending on the context. Typically, a letter represents them, and it stands in for a numerical value. In algebra, a variable represents an unknown value that you need to find. For mathematical functions and equations, you input their values to calculate the output. In an equation, a coefficient is a fixed value by which you multiply the variable.
In statistics, a variable is a characteristic of interest that you measure, record, and analyze. Statisticians understand them by defining the type of information they record and their role in an experiment or study. [Read more…] about What is a Variable?
Kurtosis is a statistic that measures the extent to which a distribution contains outliers. It assesses the propensity of a distribution to have extreme values within its tails. There are three kinds of kurtosis: leptokurtic, platykurtic, and mesokurtic. Statisticians define these types relative to the normal distribution. Higher kurtosis values indicate that the distribution has more outliers falling relatively far from the mean. Distributions with smaller values have a lower tendency for producing extreme values. When you’re assessing a sample, outliers have the greatest impact on this statistic. [Read more…] about Kurtosis: Definition, Leptokurtic & Platykurtic