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So we've learned about histograms and frequency distribution tables. Now, it's time to learn about the shape of the data and what it represents.

The Center and Spread

The first two are simple: center and spread. The center is where the data can be divided in half. The spread is the range of the data from the smallest to highest.

In later lessons, the center will be referred to as the median. We're learn more about spread when we learn about variance and standard deviation.

For an example, let's look at the following histogram:

Finding the center isn't always going to be as easy as looking at the midpoint of the $x$-axis. This is because the larger the frequency of a bin, the more "weight" you have to add to it.

A good tactic is to add up all the frequencies: $$2+4+6+3+1=16$$ And then divide that in half. $$\frac{16}2=8$$ Now, we add up the frequencies from left to right, stopping when we reach $8$. This is referred to as taking the cumulative frequency. Whatever bin you're in is the center.

Starting at bin $1$ we have a cumulative frequency of $2$. Moving to bin $2$ we have a cumulative frequency of $6$. Then, moving to bin $3$ goes to $10$. Since this is over $8$, we know the center is at bin $3$.

The spread is from $1$ (lowest point) to $5$ (highest point).

Clusters and Gaps

Next up, we have clusters and gaps. Clusters are subgroups the data could be broken into. Gaps are areas where there aren't any values.

Suppose we asked people ranging from $5$ years old to $25$ years old if they liked Pokemon. Then, we construct a histogram looking at the frequency of those who like Pokemon. We might see different clusters in the data set depending on if it was a kid, a teen, or an adult. If no one in an age range liked Pokemon, we would get a gap.

For a visual representation, let's look at the following histogram:

This graph has $3$ clusters. The first is from $0-0$. The second is from $20-30$. The third is from $30-40$. The reason $20-40$ is two different clusters is because the data from $20-30$ has similar frequencies and the data from $30-40$ has similar frequencies.

From $10-20$ we have a gap. There is no data in this section.

Outliers and Modes

An outlier is a data point that is an extreme, and is usually found excluded from the rest of the data. An outlier can be the result of an error or it could be something that naturally came about.

Suppose you're using some machine to collecting data. While it's running, Jim, who sucks, knocks into the machine and disrupts it. Congratulations, you now have an outlier. All your data might be grouped together except for a single data point at a far end which was when Jim disrupted the machine.

A mode is a major peak of a distribution. In other words, the interval with the highest frequency, or the interval that appears most often. There can be multiple modes in a distribution. If there's one mode we say it's unimodal. If there are two modes we say it's bimodal.

Let's look at the following histogram:

First up, we can see this histogram has an outlier at bin $1$. This data point is isolated from the rest and has a small frequency.

Next, we look at modes. You might be tempted to say there is one mode (unimodal), but there are actually two (bimodal). The first mode is at bin $13$ since it is the highest. The second mode is at bin $16$. This is because this bin is in it's own cluster and it's the highest for that cluster.

This type of data can also be represented in a bar graph or a pie graph, depending on the type of data you have. For example, if you have qualitative data that is based off of percents, then you may want to use a pie graph, because that type of graph shows parts of a whole or total number.

If you're just trying to represent a number of observations, a bar graph or histogram would be a good fit.

When deciding on types of graphs, it's not that they're mutually exclusive. You could have the same data that could be represented in different formats of graphs.

Shape

Finally, we have the shape of this distribution to look at. The first we look at is the symmetry. If the two halves of the distribution are mirror images, we say it's symmetric. If not, it's asymmetric.

Next we have the skew. A distribution can be normal, skewed to the left, or skewed to the right.

A normal distribution has bell-shaped curve. It represents a group of similar data. An example would be the heights of people. Most of the population would be around a similar middle height. As you move to smaller heights or taller heights, it drops off and becomes less frequent.

A distribution that is skewed to the left, (also known as a negative skew) has a tail going towards the left of the graph (towards the negative numbers), with a hump on the right. A histogram of people diagnosed with dementia based on age would have a negative skew since there would be less cases in younger people and more cases in older people.

A distribution that is skewed to the right, (also known as a positive skew) has a tail going towards the right of the graph (towards the positive numbers), with a hump on the left. A histogram of people watching Saturday morning cartoons based on age would have a positive skew since there would be more kids watched and less adults.

The final shape to look at is a uniform distribution. A uniform distribution is where all the data points are the same. It creates a horizontal line.

So that was a lot of information, but it's finally over. Now, let's try some practice problems.

You can't stop watching Shrek 2! You decide to record your age every time you watch it. Turning it into a frequency table, you get:

$$\begin{array}{|c|c|} \hline \text{age}&\text{frequency}\\ \hline 13&8\\ 14&8\\ 15&16\\ 16&15\\ 17&17\\ 18&21\\ 19&20\\ 20&15\\ 21&10\\ 22&10\\ 23&7\\ 24&5\\ 25&3\\ 26&3\\ 27&2\\ 28&1\\ 29&1\\ 30&1\\ 55&1\\ \hline \end{array}$$

First, turn this into a histogram:

Frequency Distributions

Before we talk about characteristics of frequency distributions, it might be helpful to review the basics of frequency, frequency tables, and histograms.

Although frequency tables are helpful, it is usually easiest to see how frequencies are distributed (spread out) using a histogram.

Let's talk about the characteristics of a frequency distribution:

Shape

Is the distribution (i.e., the histogram) symmetric or asymmetric?

Is the distribution unimodal or bimodal? (A mode is the most used data point. In a frequency distribution, the mode is the tallest bar in the histogram).

Unimodal = one peak in the histogram

Bimodal = two peaks in the histogram

You might also see frequency distributions represented as lines like the ones above. Imagine these lines curving over the tops of a histogram. It's still the same information.

All three are unimodal! They all have one highest point!

(Video) Frequency Distributions

by educator.com

This video by Educator.com has a explains common types of frequency distributions.

$\large{\text{Summary}}$

This video goes into further detail about frequency distributions and their features, specifically shape. She describes shapes of data: uniform, unimodal, and asymmetric.

Uniform Data: If you squint at a graph, and it seems that there’s almost a flat line, at the top, that is uniform data. No one bin is higher than the others (by much). An easy way to remember this is that uniform is all the same (like school uniforms), and uniform data all have the same height.

Unimodal Data: No matter how much you squint at the graph, there is a clear “winner”, meaning there is one spot where the data is highest, or has the highest frequency. She also talks a little bit about being “skewed right”, which will be learned more in Algebra $2$.

Asymmetric Data: If a line of symmetry is drawn at the mode, and we imagined folding the distribution on itself, we would not have $2$ sides that match up