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Statistics By Jim

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How to Find the P value: Process and Calculations

By Jim Frost Leave a Comment

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

Filed Under: Hypothesis Testing

Sampling Methods: Different Types in Research

By Jim Frost 2 Comments

What Are Sampling Methods?

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

Filed Under: Basics Tagged With: conceptual, experimental design, sampling methods

Beta Distribution: Uses, Parameters & Examples

By Jim Frost Leave a Comment

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

Filed Under: Probability Tagged With: conceptual, distributions, graphs

Geometric Distribution: Uses, Calculator & Formula

By Jim Frost Leave a Comment

What is a Geometric Distribution?

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

Filed Under: Probability Tagged With: distributions, graphs

What is Power in Statistics?

By Jim Frost 1 Comment

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?

Filed Under: Hypothesis Testing Tagged With: conceptual

Conditional Distribution: Definition & Finding

By Jim Frost Leave a Comment

What is a Conditional Distribution?

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

Filed Under: Basics Tagged With: conceptual, distributions

Marginal Distribution: Definition & Finding

By Jim Frost Leave a Comment

What is a Marginal Distribution?

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

Filed Under: Basics Tagged With: conceptual, distributions

Content Validity: Definition, Examples & Measuring

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What is Content Validity?

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

Filed Under: Basics Tagged With: conceptual

Parameter vs Statistic: Examples & Differences

By Jim Frost Leave a Comment

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

Filed Under: Basics

Spurious Correlation: Definition, Examples & Detecting

By Jim Frost 4 Comments

What is a Spurious Correlation?

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

Filed Under: Basics Tagged With: conceptual

Contingency Table: Definition, Examples & Interpreting

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What is a Contingency Table?

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

Filed Under: Basics Tagged With: conceptual, distributions

Permutation vs Combination: Differences & Examples

By Jim Frost 3 Comments

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.

  • Permutations: The order of outcomes matters.
  • Combinations: The order does not matter.

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

Filed Under: Probability Tagged With: conceptual

Cumulative Frequency: Finding & Interpreting

By Jim Frost Leave a Comment

What is Cumulative Frequency?

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.

  • Frequency: The number of times a value occurs in a dataset. For example, there are 12 4th graders in the school.
  • Frequency distribution: A table that lists all values in the dataset and how many times each one occurs. Learn more about Frequency Tables.

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.

How to Find Cumulative Frequency

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.

Construct the Cumulative Frequency Distribution for Discrete 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.

Table that displays the cumulative frequency distribution for grade level.

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.

Construct the Cumulative Frequency Distribution for Continuous Data

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.

Example of the height data.

However, you can obtain meaningful information by grouping the values into ranges and finding the frequency for each class, as shown below.

Frequency table of heights.

Then, to create the cumulative frequency table, sum each row with all preceding rows just as we did for the discrete data example.

Table that displays a cumulative frequency distribution for height data.

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.

Less Than vs. Greater Than Forms of the Table

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.

Table that displays a great than distribution.

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.

Using Graphs

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.

Bar chart displaying a cumulative frequency line.

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!

Filed Under: Basics Tagged With: conceptual, distributions

Chi-Square Goodness of Fit Test: Uses & Examples

By Jim Frost 2 Comments

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

Filed Under: Hypothesis Testing Tagged With: analysis example, conceptual, distributions, interpreting results

Sampling Error: Definition, Sources & Minimizing

By Jim Frost 4 Comments

What is Sampling Error?

Sampling error is the difference between a sample statistic and the population parameter it estimates. It is a crucial consideration in inferential statistics where you use a sample to estimate the properties of an entire population. [Read more…] about Sampling Error: Definition, Sources & Minimizing

Filed Under: Hypothesis Testing Tagged With: conceptual

Cohort Study: Definition, Benefits & Examples

By Jim Frost Leave a Comment

What is a Cohort Study?

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

Filed Under: Basics Tagged With: conceptual, experimental design

Inter-Rater Reliability: Definition, Examples & Assessing

By Jim Frost Leave a Comment

What is Inter-Rater Reliability?

Inter-rater reliability measures the agreement between subjective ratings by multiple raters, inspectors, judges, or appraisers. It answers the question, is the rating system consistent? High inter-rater reliability indicates that multiple raters’ ratings for the same item are consistent. Conversely, low reliability means they are inconsistent. [Read more…] about Inter-Rater Reliability: Definition, Examples & Assessing

Filed Under: Hypothesis Testing Tagged With: analysis example, conceptual, interpreting results

How to Find the Mode

By Jim Frost Leave a Comment

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

Filed Under: Basics Tagged With: conceptual

Bimodal Distribution: Definition, Examples & Analysis

By Jim Frost 1 Comment

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

Filed Under: Basics Tagged With: conceptual, distributions, graphs

Margin of Error: Formula and Interpreting

By Jim Frost Leave a Comment

What is the Margin of Error?

The margin of error (MOE) for a survey tells you how near you can expect the survey results to be to the correct population value. For example, a survey indicates that 72% of respondents favor Brand A over Brand B with a 3% margin of error. In this case, the actual population percentage that prefers Brand A likely falls within the range of 72% ± 3%, or 69 – 75%. [Read more…] about Margin of Error: Formula and Interpreting

Filed Under: Hypothesis Testing Tagged With: conceptual, interpreting results

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