Descriptive Statistics and Frequency Distributions, Explained

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Descriptive Statistics and Frequency Distributions

Research Methods in Psychology · Chapter 14: Descriptive StatisticsSection 14.1 · lesson 84 of 949 min · free with a free account

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In this lesson

Turning a pile of numbers into something a reader can understand.

  • Distinguish descriptive from inferential statistics
  • Construct a frequency distribution from raw data
  • Select the right display for a given variable type
  • Identify symmetric, skewed and bimodal shapes

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Descriptive Statistics

Descriptive Statistics and Frequency Distributions

Raw data is unreadable. That is not a figure of speech. A column of two hundred numbers tells a human being almost nothing. This section is about the first step that makes data legible. By the end you will distinguish descriptive from inferential statistics. And explain the role each one plays in research. You will construct a frequency distribution from raw data, and present it in a table. You will select an appropriate graphical display for a given variable. A bar chart, a histogram, or a frequency polygon. You will identify common shapes, including symmetric, skewed, and bimodal forms. And you will see how the choice of bin width shapes the picture you present.

Two Activities

Why Description Comes First

Statistical work in research splits into two activities. Descriptive statistics summarize the dataset you have in hand. How high the scores are on average. How widely they vary. What the distribution looks like overall. Inferential statistics use that summary to reach beyond the data. They draw conclusions about the larger population the data were sampled from. Whether the observed differences would generalize. Whether a hypothesis is supported. Both activities matter, but description always comes first. A researcher who runs inferential tests without inspecting the data is working blind. And many of the most embarrassing errors in published research begin exactly there.

The Foundational Tool

The Frequency Distribution

The foundational tool of descriptive analysis is the frequency distribution. It is a table or graph recording how often each value occurs in the data. Or how often each range of values occurs. Make that concrete with a class of two hundred students. They have just completed a final exam. The instructor has raw scores on her screen, running from the low forties into the high nineties. In no particular order, with no visible pattern. Before any interpretation can begin, those numbers need to be organized. The frequency distribution puts them into a form human eyes can read. Once you can see the shape, the later decisions follow naturally.

In a Table

Bins and the Judgment They Require

Start with the tabular version. A simple frequency distribution lists each distinct value, with the number of times it appears. For a small set of categories, that works beautifully. For continuous variables, or large datasets, it produces an unwieldy table. Two hundred exam scores could easily give you sixty distinct values. So the researcher groups values into bins of equal width. Then counts how many observations fall into each bin. Choosing the bin width is a judgment call, and it has real consequences. Bins that are too narrow simply reproduce the noise of the raw data. Bins that are too wide blur away meaningful structure. A useful starting point is roughly seven to fifteen bins across the range. Then adjust, based on what the resulting distribution reveals.

In a Picture

Histograms and Frequency Polygons

The histogram is the visual counterpart to the grouped frequency table. Each bin becomes a bar. The height of that bar represents the count of observations in the bin. One detail of the drawing carries real meaning. The bars are drawn adjacent to one another, with no gaps between them. That signals that the underlying variable is continuous. And it is exactly what distinguishes a histogram from a bar chart. Bar charts are for categorical data, and their bars are separated. There is a third option worth knowing, the frequency polygon. It connects the midpoints of histogram bars with line segments. That makes it particularly useful for comparing two distributions on the same axes.

Same Data, Different Read

Table or Graph

A grouped frequency table and a histogram contain the same numerical information. Exactly the same information. Nothing is added by drawing it. And yet they are not equally easy to read. The histogram reveals shape at a glance. The location of the peak. The symmetry of the tails. The presence of unusual gaps that nobody was looking for. The table preserves the exact counts, which the histogram approximates. But it requires the reader to construct the shape mentally, row by row. Both have their uses, and a paper often includes both. For inspection, though, the visual nearly always wins.

Shape

What Summary Statistics Cannot Tell You

Now think about why shape matters at all. The shape of a distribution carries information summary statistics cannot. A mean and a standard deviation are two numbers. Many very different datasets produce the same two numbers. Three features deserve particular attention when you look at a distribution. Where the distribution is concentrated. How it tails off on either side. And whether there are unusual features, like multiple peaks or extreme outliers. Three common shapes recur often enough to have names. Symmetric, skewed, and bimodal.

Shape One

Symmetric Distributions

A symmetric distribution looks roughly the same on both sides of its center. The well-known bell curve of the normal distribution is the most familiar example. But here is a point students routinely get wrong. Symmetric does not mean bell-shaped. Symmetric distributions can be flat, peaked, or irregular. The only requirement is that the two halves are mirror images. Many measured psychological variables approximate symmetric forms. Particularly those produced by averaging across many separate influences. That is not a coincidence, and it becomes important later in the course.

Shape Two

Skewed Distributions

A skewed distribution has one tail considerably longer than the other. Positive skew means a long tail extending toward larger values. That is typical of income, and of reaction times. And of many measures bounded at zero on one end, but unbounded on the other. You cannot respond in less than no time, but you can always take longer. Negative skew is the mirror image, with a long tail toward smaller values. That is typical of test scores when most students perform well and a few struggle. Notice that the name refers to the tail, not to the bulk of the data. Recognizing skew matters for a practical reason. It influences which summary statistics are appropriate. The next section takes that question up directly.

Shape Three

Bimodal and Multimodal Distributions

Some distributions show two or more distinct peaks. Those are bimodal, or more generally multimodal. And they often carry a specific message about the data. They indicate that the data combine two underlying populations. Male and female heights, pooled together, is the classic illustration. Or scores on a task that some participants perform carefully and others perform inattentively. Now think about what a single average would describe there. Summary statistics across the whole dataset describe a hybrid. A hybrid that may not represent either subgroup well, or even at all. So identifying and analyzing the subgroups separately is often the better approach.

Matching Display to Variable

Five Cases

Different kinds of variable call for different displays. It is a small choice, and it shapes what readers perceive at a glance. For categorical variables, use a bar chart with separated bars. The categories have no inherent order, and the gaps emphasize that they are discrete. For ordinal variables with few levels, also a bar chart. Arranged by category order, because order matters even though distances are not numerical. For continuous or interval variables, a histogram or frequency polygon. Adjacent bars signal the underlying continuity, and reveal shape clearly. For comparing two distributions, overlay frequency polygons, or use back-to-back histograms. Shared axes make differences in shape and center visible immediately. And for very small datasets, under about thirty observations, avoid binning altogether. A dot plot or stem and leaf display preserves every individual value.

Honesty and Next Steps

Bin Width, and Where This Leads

One warning before moving on, about a choice that looks purely cosmetic. Two histograms of the same data, with different bin widths, can look strikingly different. Wide bins smooth over local features, and emphasize the overall shape. Narrow bins reveal fine structure, but can also look noisy and irregular. There is no single correct bin width. The right choice depends on what you want the reader to see. Which is exactly why you should report it. Reporting the bin width, and sometimes showing the data at more than one resolution, is honest description. From here, three families of summary statistics follow. Central tendency, the mean, median, and mode, describing where the distribution is centered. Variability, the range, variance, and standard deviation, describing how widely data spread. And position, percentiles and z scores, locating an individual score within the whole.

About Research Methods in Psychology at UPI Study

Research Methods in Psychology (Psychology 111) is a self-paced online course of 94 video lessons in 16 chapters, from the scientific method and research ethics to designs, data collection, sampling, measurement, validity, experiments, descriptive statistics and hypothesis testing. It carries a NCCRS credit recommendation for 3 semester credits.

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Research Methods in Psychology carries a NCCRS credit recommendation. These universities work with NCCRS; some are UPI Study's signed partners. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.

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FAQ

Questions students ask us

About Descriptive Statistics and Frequency Distributions

What is the difference between descriptive and inferential statistics?
Descriptive statistics summarize the data in hand, such as how high scores are on average, how much they vary, and the shape of the distribution. Inferential statistics use those summaries to draw conclusions about the larger population the sample came from, such as whether a difference would generalize or whether a hypothesis is supported. Description comes first, because running inferential tests without inspecting the data invites errors.
What is a frequency distribution?
A frequency distribution is a table or graph showing how often each value, or each range of values, occurs in a dataset. When there are many distinct values, such as 200 exam scores, values are grouped into bins of equal width and the observations in each bin are counted. Bin width is a judgment call: bins that are too narrow reproduce noise, and bins that are too wide hide structure.
What is the difference between a histogram and a bar chart?
A histogram displays a continuous variable with adjacent bars and no gaps, while a bar chart displays categorical data with separated bars. In a histogram, each bar is a bin and its height is the count of observations in that bin. A frequency polygon connects the midpoints of the histogram bars with lines, which makes it useful for comparing two distributions on the same axes.
What is the difference between positive and negative skew?
A positively skewed distribution has a long tail toward larger values, and a negatively skewed distribution has a long tail toward smaller values. Reaction times and income are typically positively skewed, since they cannot fall below zero but can stretch far upward. Test scores are often negatively skewed when most students do well and a few struggle. The name refers to the direction of the tail, not to where most of the data sit.

About the Research Methods in Psychology course

How many college credits is Research Methods in Psychology?
Research Methods in Psychology (Psychology 111) carries a NCCRS credit recommendation for 3 semester credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
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