# Histograms

March 22, 2019

# What is a histogram?

A histogram graphically summarizes the distribution of numerical data. Typically, it consists of vertical rectangular bars which show the frequency of data (along y-axis) in successive class intervals (x-axis) of equal size. The height of each bar of a histogram is proportional to the frequency of the data.

Let’s take a sample data of heights (in cm) of 50 students in a class.

Heights (in cm) of students in a class

For plotting histogram, the first step is to divide the entire range of values into a series of intervals or bins. Here the heights range from 150.6 cm to 189.1 cm. Let’s divide the range into 8 intervals, each of width 5. The frequency distribution table is:

Frequency distribution table for the above data

Each row of the table tells us the number of data points in that range. For example, the number of students with heights in range 180 - 185 is 4 (the students’ heights are 182.1, 180.9, 181.4 and 181.9).

Here’s the histogram for this data:

We can see that the histogram visualizes the distribution of data over a continuous interval. The height of each bar in the histogram represents the frequency at each interval / bin. For example, as we discussed earlier, there are 4 students whose heights are in the range 180 - 185.

# Creating histogram with Python

Now let’s see how we would create the histogram with Python. We will be using the `matplotlib`

library, which provides a `hist()`

function for plotting histograms. The two most important parameters the function takes are

`range`

— The lower and upper range of the bins (defaults to minimum and maximum value in the data). If any of the data values are outside the range specified, they will be ignored.`bins`

— The number of bins/intervals to divide the range into

Let’s see the code in action:

```
import matplotlib.pyplot as plt
# height data
heights = [
158.9, 175.4, 167.2, 162.3, 152.6, 159.2, 176.9, 155.8, 163.6, 181.9,
167.4, 161.5, 180.9, 171.6, 165.6, 160.1, 172.7, 155.3, 171.2, 162.7,
175.7, 182.1, 188.4, 157.0, 189.1, 168.7, 178.9, 162.3, 186.9, 178.8,
166.0, 165.9, 169.3, 155.4, 178.9, 181.4, 162.2, 176.5, 165.9, 150.6,
172.7, 174.9, 150.9, 152.2, 172.6, 170.7, 171.5, 169.4, 155.9, 162.7,
]
# plot histogram
# 'ec' stands for edge color. we add black borders to the bars.
plt.hist(heights, bins=8, range=(150,190), ec='black')
# add title and axis labels
plt.title("Histogram – Height of Students")
plt.ylabel("Frequency")
plt.xlabel("Height Bins")
# display the plot
plt.show()
```

Notice how we choose `range`

and `bins`

such that all the intervals have integer values — in this case, 150 - 155, 155 - 160, and so on.

Note: We should choose thenumber of binswisely for histograms. Way too few bins does not reveal any information, since all the data points will belong to the same bar. On the other hand, too many bins will make a separate bar for each data point. As a rule of thumb, for realistic data you usually want around 20 - 50 bins.

# Histograms for various types of frequency distributions

The shape of the histogram reflects the underlying distribution of data. Hence, often taking a single glance at the histogram allows us to easily understand the data distribution. Below, we will discuss some distributions which are encountered commonly and their corresponding histogram shapes.

# Skew

## Symmetric

In a symmetric distribution, data points are symmetrically distributed around the *mean*. In these cases, the mean is approximately equal to the median. An example of a symmetric distribution is the height of individuals in a locality. A lot of the data observed in nature follows this distribution.

The histogram representing a symmetric distribution has the left and right hand sides roughly equally balanced around the centre. You can observe that the left and right tails in the graph below have about the same length.

Histogram for data with symmetric distribution

## Right Skewed (Positively skewed)

Variables representing income or wealth of individuals etc., are right (positively) skewed. This is because the majority of individual have either low income or medium income, but a small minority of individuals are very rich or extremely rich.

The histogram of a right skewed distribution has a long tail to the right. In these cases, the mean is typically greater than the median.

Histogram for right-skewed data

## Left Skewed (Negatively skewed)

Similarly, if the histogram has a long tail to the left, we can assume that the data is from a negatively skewed distribution. In a left skewed distribution, most data points will be greater than the mean.

Observe the shape of the histogram with a long left tail. In this case, the mean is typically less than the median.

Histogram for left-skewed data

# Kurtosis

Apart from the left / right skew of the data distribution, another important thing to notice in histograms is the “tailedness” of a distribution, or how much of the data is close to the center vs in the tails. This property of the distribution is known as its **kurtosis**.

## Mesokurtic (Normal tailed)

To measure how heavy or light-tailed a distribution is, we take the **normal distribution** (also known as **Gaussian distribution**) as the point of comparison. Below is a histogram plot of data points from a normal distribution.

Histogram for data from normal distribution

## Leptokurtic (Heavy tailed)

A distribution whose tails are heavier than the normal distribution is called **leptokurtic**. Often, this kind of distribution will have heavy tails as well as a taller center, but lesser data points at a moderate distance from the center.

Histogram for data from a heavy-tailed distribution

## Platykurtic (Light tailed)

A distribution whose tails are thinner than the normal distribution is called **platykurtic**. In some cases (as in the plot below), the tails might be completely non-existent.

Histogram for data from a distribution with non-existent tails

# Histogram for multiple distributions

Histograms are also useful for comparing different subsets of the data.

Let’s say we have data containing the salary (in $k) of 1000 employees in USA and UK. We will compare their distribution by plotting two histograms simultaneously in the same graph within the same axes.

Let’s first create the data which we will be plotting:

```
# random seed so that the generated values are always same
np.random.seed(2)
# 1000 values for USA incomes with mean 100 and standard deviation 30
income_USA = 30 * np.random.randn(1000) + 100
# 1000 values for UK incomes with mean 85 and standard deviation 20
income_UK = 20 * np.random.randn(1000) + 85
```

## Superimposed histogram

In a superimposed histogram, we display the histograms layered on top of each other.

Here’s the code to create a superimposed histogram:

```
# Superimposed histogram
plt.hist([income_USA, income_UK], label=['USA', 'UK'], histtype='stepfilled',
alpha=0.6, bins=30, ec='black')
# 'alpha' sets the 'opacity'.
# if alpha = 1.0 (default) we won't be able to see the second histogram behind.
# add title and axis labels
plt.title("Histogram – Height of Incomes")
plt.ylabel("Frequency")
plt.xlabel("Income Bins")
# add legend
plt.legend(loc='upper right')
# display plot
plt.show()
```

Notice some things about the code above:

- To plot multiple data together, we pass all of them to the
`hist()`

function. - We use the
`label`

argument to label the data, and the`legend()`

function to display the labels. - To create a superimposed histogram, we set the parameter
`histtype`

’s value to be`'stepfilled'`

. There are other histogram types as well, which we will see in the next section. - Lastly, since the histograms are layered on top of each other, we used the
`alpha`

parameter to make the histograms a little transparent so that we can also see ‘behind’ it.

Here’s the resulting plot:

From the histogram, we can see that on average income levels in USA are higher than in UK (note that income is on the x-axis, so higher income means the histogram is shifted to the *right*).

We can also see that incomes of employees from USA have a higher spread (since the histogram is wider).

## Stacked Histogram

Apart from superimposing the plots, there are other ways to plot the distributions in the same graph. One option is to *stack* the multiple sets of data over one another. For that, we assign the parameter `histtype`

the value `'barstacked'`

.

```
# Stacked histogram
plt.hist([income_USA,income_UK], label=['USA','UK'], histtype='barstacked', bins=30, ec='black')
plt.title("Histogram – Height of Incomes")
plt.ylabel("Frequency")
plt.xlabel("Income Bins")
plt.legend(loc='upper right')
plt.show()
```

Stacking makes it easier to see the combined frequency, while making it harder to see the exact frequency within each category.

## Side by side histogram

Finally, we can also plot the data side by side. To achieve this, we set the parameter `histtype`

to `'bar'`

. This plot makes it easier to compare the data frequency for each class interval.

```
# Side by side histogram
plt.hist([income_USA,income_UK], label=['USA','UK'], histtype='bar', bins=20, ec='black')
plt.title("Histogram – Height of Incomes")
plt.ylabel("Frequency")
plt.xlabel("Income Bins")
plt.legend(loc='upper right')
plt.show()
```

We can visually check the frequencies of income levels in USA and UK, and determine where one dominates the other. From the above plot, we can see that at very low and very high income levels, USA has a higher frequency, whereas in the middle regions, UK has a higher frequency.

# Summary

**Histograms**summarize large data sets graphically- It is easy to understand the underlying
**distribution**by looking at the histogram — for example, we can infer whether the distribution is symmetrical or skewed. - Multiple distributions can be easily compared by plotting a
**superimposed histogram**. - You saw how to plot histograms using
`matplotlib`

’s`hist()`

function, and the various parameters it supports such as`bins`

,`range`

,`ec`

(for adding borders to the bars),`label`

,`alpha`

(for transparency) and`histtype`

.

# References

We covered all the important parameters that the `matplotlib`

histogram supports. If you would like to read more about the other parameters the `matplotlib`

histogram supports, you can check the documentation here.