What is a histogram in statistics? How does it visualize data? And how can this visualization help you with data analysis?
In this video, I’ll show you how to ace your next statistics exam and take your data analysis to the next level using histograms.
Histograms are a standard tool in statistics and are essential for many academic papers. To help you understand and use histograms effectively, I’ll walk you through the basics today.
Of course, I’ll also show you how to create a histogram for any dataset in no time.
1. What is a Histogram?
A histogram is a type of chart that represents a frequency distribution. As you can see in the graphic, the x-axis represents intervals, while the y-axis shows their corresponding frequencies.
A key characteristic of a histogram is that the bars are directly adjacent to one another, with no gaps in between. This emphasizes the continuous nature of the data, as each bar represents a range of values rather than discrete categories. This is because histograms are used for continuous data (e.g., measurements like weight, length, or time spans).
In contrast, bar charts represent categorical data (nominal data such as the number of students in different study programs like law, psychology, or business administration). That’s why bars in a bar chart are separated from each other.
It’s also crucial that the y-axis of a histogram starts at a frequency of 0. The height of each bar represents the number of data points in that interval.
If the baseline is altered, the perceived heights of the bars change, potentially distorting the actual distribution of the data. This could lead to an overestimation of low frequencies or an underestimation of high frequencies.
2. Where Are Histograms Used?
Histograms are widely used across various fields. In economics, for example, they help analyze income distribution across different demographic groups. In medicine, they assist in understanding the distribution of measurements like blood pressure or BMI within a population.
They are also crucial for fundamental statistical data analysis, such as checking whether a dataset follows a normal distribution.
3. Creating a Histogram in Statistics
Let’s create a histogram using a real-world example. We have a dataset of exam scores from the last statistics test:
53, 41, 71, 91, 99, 93, 87, 74, 97, 81, 85, 89, 78, 61, 66, 71, 86.
First, you need to create a frequency distribution table and group the scores into intervals.
The intervals must have equal width, ensuring that all bars are the same size. If the intervals are too wide, important details might be lost, whereas too narrow intervals could make the chart too complex. For this example, I’ve chosen intervals of 10 points each (40-49, 50-59, 60-69, etc.).
In statistics, class intervals for histograms are typically chosen so that the lower boundary is inclusive, and the upper boundary is exclusive.
This means that an interval of 60-69 includes all values from 60 up to but not including 69. If we instead used an interval of 60-70, the value 70 would belong to two intervals (both 60-70 and 70-80), leading to ambiguity. To avoid this issue and ensure a clear, unambiguous assignment of data points to intervals, histogram intervals do not overlap.
Now let’s look at the frequencies.
- One student scored in the 40-49 range.
- Another student scored between 50-59.
- Two students scored between 60-69.
- Four students scored between 70-79.
- And so on…
Now, you need to plot this data using software like Excel or R. The result for our example looks like this:
4. Understanding a Histogram in Statistics
Interpreting a histogram in statistics is a crucial step in understanding your collected data. A histogram provides a visual representation of how data is distributed.
It helps identify patterns and anomalies that may indicate specific trends or issues. Keep in mind that in density histograms, probabilities are represented by the area of the bars, while in frequency histograms, the bar height indicates the number of observations in each interval.
1. Data Distribution
Histograms show the frequency of data within different intervals, making it easy to assess distribution at a glance. Researchers can quickly determine whether the data is normally distributed, skewed left or right, or exhibits other patterns like bimodal distributions.
A normal distribution, often called a bell curve, means that most data points cluster around a central value, with symmetrical tails extending on both sides. In a university setting, this could represent exam scores, where most students achieve average marks, while very high or very low scores are less common.
A skewed distribution indicates that the data is asymmetrically spread. A positively skewed (right-skewed) histogram shows a concentration of low values with a few high values—such as the time students spend studying for a subject. Many may spend only a little time, while a few invest a lot. A negatively skewed (left-skewed) distribution suggests the opposite.
A bimodal distribution, featuring two peaks, may indicate the presence of two distinct groups. For example, in a class attended by both first-year and advanced students, two peaks might suggest that each group tends to score differently.
2. Identifying Anomalies
Visualizing data can reveal outliers, unusual patterns, or anomalies that may warrant further investigation. The width of the intervals shows how data is grouped.
- Narrow bars indicate a detailed distribution.
- Wider bars provide a more generalized overview.
- Bar height represents the number of observations in each interval—taller bars indicate higher frequencies.
3. Comparing Datasets
Histograms allow for easy comparison of two or more datasets. You can use them to examine how data is distributed under different conditions or across different groups.
4. Hypothesis Testing
Histograms can help formulate or test hypotheses about data. For example, if you hypothesize that a particular variable follows a normal distribution, a histogram in statistics can confirm or disprove this assumption.
5. Decision-Making
In practice, such as in quality control, histograms are used to determine whether a business process meets specific specifications.
5. Interpreting Our Example Histogram
To better understand a histogram in statistics, I’ll now pose a few questions about our example. Feel free to pause and try answering before checking the solutions.
- Would you say the data is symmetric, or is it skewed left or right?
You can see that the taller bars are on the left side. This suggests a left-skewed distribution, meaning the data has negative skewness. In other words, students scored relatively high in this exam.
- What is the mode of this dataset?
The mode is the interval with the highest frequency. In this case, most students scored between 80 and 89, making this the mode.
- How many students scored up to 69 points?
Adding the first three bars: 1+1+2 = 4 students scored up to 69 points.
- How many students scored at least 80 points?
Adding the last two bars: 5+4 = 9 students scored at least 80 points.
- How many students scored between 60 and 89 points?
Adding the middle bars: 2+4+5 = 11 students scored within the intervals 60-69, 70-79, and 80-89.
6. Histograms and Probabilities
Histograms help navigate large datasets. These visual representations display probability distributions, which are essential for understanding a dataset’s dynamics.
Returning to our exam example: the bar heights indicate how many students fall within specific score ranges. But they also reflect the probability of a randomly selected student achieving a particular result.
A clustering of values around a central score suggests a normal distribution, which many statistical tests assume. The histogram helps determine whether this assumption holds or if another testing approach is needed.
Histograms also allow us to infer conclusions about an entire population from a sample, provided that the sample is representative and sufficiently large. For instance, a histogram of a class’s exam scores can provide insights into the performance of all students in the program.
All in all, a histogram is like a Swiss Army knife in a statistics. If you want to dive deeper, I highly recommend Andy Field’s book Discovering Statistics.