How to Read and Interpret a Box and Whiskers Plot
What is a Box and Whiskers Plot?
Sometimes we need more elaborated details in various distributions or datasets that may not be fulfilled by the measures of any central tendency like mean, median, and mode. Information on the variability or the dispersion of the data demands a much more concrete foundation. This demand can be fulfilled by a box and whiskers plot. This may arise a predictable question that is “what is a box and a whiskers plot?” the question can be answered satisfactorily. A box plot is a graph that offers us a much firm indication or idea about how the values in the data should be spread. Box plots extend its lines from the boxes which are normally called whiskers. Whiskers are used to indicate variability outside the upper and the lower quartiles. Being non-parametric is one of the features of the Box plot. And this feature of the Box plot actually helps to display a variation of a statistical population in the samples where it does not make any assumptions about the underlying statistical distribution. The gaps between the different parts of the box indicate the degree of dispersion (spread) and the skewness present in the data, along with the show outliers. Box plots are drawn in two ways, i.e., we can choose to draw it either horizontally or vertically. A box plot is like a chart that we often use in exploratory data analysis.
Box plots have a five-number summary of a set of data that includes the minimum score, first quartile (lower), median, third quartile (upper), and the maximum score.
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Although it may seem that a box plot is primitive if compared to a histogram or a density plot, they have an advantage of occupying less space, which is quite useful when comparing distributions between many groups or datasets.
Elements of Box and Whisker Plot
The Minimum Score: The minimum score is the lowest score after excluding the outliers.
The Maximum Score: The maximum score is the highest score after excluding the outliers.
The Median: The median represents the midpoint of a data which can be shown using the line that divides the box into two halves (it is sometimes also called the second quartile). It is seen that most of the scores are much greater or equal to the value and half are less.
The Lower Quartile: We can also call the lower quartile as the first quartile which falls below 25 percent of the scores.
The Upper Quartile: The upper quartile is also known as the third quartile and it falls below 75 percent of the scores.
The Whiskers: The upper and the lower whiskers are the lines that represent the scores outside the middle 50%.
The Interquartile Range (or IQR): The interquartile range is the middle box plot that represent the scores between 25 percent to 75 percent i.e., 50 present scores.
Box and Whisker Plot Solved Examples
Question 1) Given below is a sample of the weight of 10 boxes of raisins in grams:
25, 28, 29, 29, 30, 34, 35, 35, 37, 38.
Solution
Step 1: First create a box-plot of the data and start from the smallest to the largest. But here the data is already in increasing order. So let us move on to our next step.
Step 2: Find the median in this step along with the mean of the two middle numbers.
Therefore, median = 30+34/2 = 32
Step 3: Moving on to our next step where we have to find the quartile. Our first quartile would be the median of the data points. That is to the left of the median so it’s 29. The third quartile would be at the right of the median so it’s 35.
Step 4: This is our last step and here we will be completing the five-number summary and do do this, we have to find the min and max. As we know that our min is 25 which is the smallest data point and our max is 38 which is the largest data point.
So finally, we can put our five-number summary as 25, 29, 32,35, and 38.
Question 2: The five-number summary data set that we have used in our above example, we will plot our box and whiskers plot.
25, 28, 31, 34, 37, and 40.
Solution 2:
Step 1: Start scaling and labelling the axis that will fit the summary.
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Step 2: Next we will be drawing a box from Q1 to Q3 vertically through the median. So the Q1 as we know was 29, Q3 was 35 and the median was 32.
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Step 3: Time for the whiskers to be drawn from Q1 to minimum and from Q3 to maximum. The minimum is 25 and the maximum is 38.
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And boom! our box and whiskers plot is ready.
FAQs on Box and Whiskers Plot Explained for Students
1. What is a Box and Whisker Plot in Maths?
A Box and Whisker Plot, also known as a box plot, is a standardised graphical method used in statistics to display the distribution of a dataset. It is particularly useful for showing the five-number summary: the minimum value, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum value. This visual summary helps in quickly understanding the data's central tendency, spread, and skewness.
2. What are the five essential components of a Box and Whisker Plot?
The five essential components, known as the five-number summary, provide the foundation for any Box and Whisker Plot. They are:
Minimum: The smallest value in the dataset, representing the start of the lower whisker.
First Quartile (Q1): The value below which 25% of the data falls. It marks the left edge of the box.
Median (Q2): The midpoint of the dataset. 50% of the data is below this value. It is represented by a line inside the box.
Third Quartile (Q3): The value below which 75% of the data falls. It marks the right edge of the box.
Maximum: The largest value in the dataset, representing the end of the upper whisker.
3. How do you create a Box and Whisker Plot from a given set of data?
To create a Box and Whisker Plot, you follow five main steps:
Order the Data: Arrange your dataset in ascending order, from the smallest value to the largest.
Find the Median (Q2): Identify the middle value of the dataset. This is your second quartile (Q2).
Find the Quartiles (Q1 and Q3): Find the median of the lower half of the data (this is Q1) and the median of the upper half of the data (this is Q3).
Identify Extremes: Determine the minimum and maximum values in your dataset.
Draw the Plot: Draw a number line that covers the range of your data. Draw a box from Q1 to Q3, with a line inside at the Median (Q2). Then, draw 'whiskers' from the box out to the minimum and maximum values.
4. How does the 'box' in a Box and Whisker Plot relate to the Interquartile Range (IQR)?
The 'box' visually represents the Interquartile Range (IQR), which is a key measure of statistical dispersion. The left side of the box is the first quartile (Q1), and the right side is the third quartile (Q3). Therefore, the width of the box itself (Q3 - Q1) is the IQR. This box contains the middle 50% of the data, providing a clear visual of the data's central spread and helping to ignore the influence of extreme outliers.
5. What does a skewed Box and Whisker Plot indicate about the data?
The shape of a box plot reveals the skewness of the data distribution:
Symmetrical Distribution: If the median line is in the exact centre of the box and the whiskers are of equal length, the data is likely symmetrical.
Positive Skew (Skewed Right): If the median is closer to the first quartile (Q1) and the whisker on the right side is longer, the data is positively skewed. This means the tail of the distribution extends towards higher values.
Negative Skew (Skewed Left): If the median is closer to the third quartile (Q3) and the whisker on the left side is longer, the data is negatively skewed, indicating a tail towards lower values.
6. What key features should you focus on when comparing two different Box and Whisker Plots?
When comparing two box plots, focus on three key aspects:
Difference in Median: Compare the positions of the median lines (Q2) to see which dataset has a higher or lower central tendency.
Difference in Spread: Compare the widths of the boxes (the IQRs). A wider box indicates a greater spread or variability in the middle 50% of the data. Also, compare the total length of the whiskers to understand the overall range.
Difference in Skewness: Observe the position of the median within each box and the lengths of the whiskers to determine if the datasets have similar or different skews.
7. When is it better to use a Box and Whisker Plot instead of a Histogram?
While both are useful, a Box and Whisker Plot is generally better for comparing the distributions of several datasets at once, as they can be easily drawn side-by-side on the same scale. It excels at highlighting medians, quartiles, and outliers for comparative analysis. In contrast, a histogram is more effective for understanding the detailed shape of a single dataset's distribution, such as identifying if it is bimodal (has two peaks), which a box plot cannot show.
8. How are outliers identified and represented on a Box and Whisker Plot?
In a modified Box and Whisker Plot, outliers are identified using the Interquartile Range (IQR). A common rule is to classify a data point as an outlier if it falls more than 1.5 times the IQR below the first quartile (Q1) or more than 1.5 times the IQR above the third quartile (Q3). These outliers are then plotted as individual points (e.g., dots or asterisks) on the graph. The 'whisker' in this case does not extend to the outlier, but rather to the highest or lowest data point within the 1.5 x IQR range.
