Unit 8 Data Sets And Distributions — Unit Plan

A dot plot is a way to show data on a number line. It is the same idea as the line plot from Grade 4 — just with dots instead of X marks.

Every dot plot has four parts:

• Title — what the data is about (for example, “Hours on Homework”).
• Key — what each dot stands for (usually each dot = 1 student).
• Axis label — what the numbers along the bottom mean.
• Dots — one dot for each piece of data, stacked above the matching number.

To read a dot plot, count the dots above a single number. That count is the frequency for that value. The number with the tallest stack is the mode — the most common value.

To build a dot plot from a frequency table, draw the right number of dots above each tick to match the frequency in the table. The total number of dots equals the total in the table.

We often collect and analyze data because we are interested in learning what is “typical,” or what is common and can be expected in a group.

Sometimes it is easy to tell what a typical member of the group is. For example, we can say that a typical shape in this set is a large circle.

Just looking at the members of a group doesn’t always tell us what is typical, however. For example, if we are interested in the side length typical of squares in this set, it isn’t easy to do so just by studying the set visually.

In a situation like this, it is helpful to gather the side lengths of the squares in the set and look at their distribution, as shown in this dot plot.

A dot plot for "side lengths in centimeters". The numbers 1 through 8 are indicated. The data are as follows: 2 centimeters, 4 dots. 3 centimeters, 5 dots. 4 centimeters, 3 dots. 5 centimeters, 3 dots. 6 centimeters, 2 dots. 7 centimeters, 1 dot.

We can see that squares with 3 centimeter sides are the most common and many others are about the same size. That means we could say that side lengths of about 3 centimeters are typical of squares in this set.

A histogram shows how often values fall into different ranges. Each bar covers a bin — a range of numbers like “20 up to (but not including) 30”, written [20, 30).

• The height of a bar tells you how many values fall in its bin.
• Each value belongs to exactly one bin. The left number is included; the right number is not. So 30 goes in [30, 40), not in [20, 30).
• The total number of values is the sum of all the bar heights.

To build a histogram from a categorized frequency table, draw one bar per row, with the bar's height matching the frequency in that row. Put the bars next to each other — they touch, because the intervals are continuous.

A histogram looks a lot like the bar graph from 2nd grade, but a histogram is for numbers, not categories. That’s why the bars touch.

In addition to using dot plots, we can also represent distributions of numerical data using histograms.

Here is a dot plot that shows the weights, in kilograms, of 30 dogs, followed by a histogram that shows the same distribution.

A dot plot, the numbers 10 through 35, in increments of 5, are indicated. The 30 data values are as follows: 10 kilograms, 1 dot. 11 kilograms, 1 dot. 12 kilograms, 2 dots. 13 kilograms, 1 dot. 15 kilograms, 1 dot. 16 kilograms, 2 dots. 17 kilograms, 1 dot. 18 kilograms, 2 dots. 19 kilograms, 1 dot. 20 kilograms, 3 dots. 21 kilograms, 1 dot. 22 kilograms, 3 dots. 23 kilograms, 1 dot. 24 kilograms, 2 dots. 26 kilograms, 2 dots. 28 kilograms, 1 dot. 30 kilograms, 1 dot. 32 kilograms, 2 dots. 34 kilograms, 2 dots.

A histogram, the horizontal axis is labeled “dog weights in kilograms” and the numbers 10 through 35, in increments of 5, are indicated. On the vertical axis the numbers 0 through 10, in increments of 2, are indicated. The data represented by the bars are as follows: Weight from 10 up to 15, 5. Weight from 15 up to 20, 7. Weight from 20 up to 25, 10. Weight from 25 up to 30, 3. Weight from 30 up to 35, 5.

In a histogram, data values are placed in groups, or “bins,” of a certain size, and each group is represented with a bar. The height of the bar tells us the frequency for that group.

For example, the height of the tallest bar is 10, and the bar represents weights from 20 to less than 25 kilograms, so there are 10 dogs whose weights fall in that group. Similarly, there are 3 dogs that weigh anywhere from 25 to less than 30 kilograms.

Notice that the histogram and the dot plot have a similar shape. The dot plot has the advantage of showing all of the data values, but the histogram is easier to draw and to interpret when there are a lot of values or when the values are all different.

Here is a dot plot showing the weight distribution of 40 dogs. The weights were measured to the nearest 0.1 kilogram instead of the nearest kilogram.

A dot plot for “dog weights in kilograms”. The numbers 8 through 36, in increments of 2, are indicated. There is 1 dot on each of the following values: 10 kilograms,, 11 kilograms, 11.3 kilograms, 12 kilograms, 12.1 kilograms, 13 kilograms, 14.7 kilograms, 15 kilograms, 15.1 kilograms, 16 kilograms, 16.5 kilograms, 17 kilograms, 18 kilograms, 18.5 kilograms, 19 kilograms, 19.1 kilograms, 20 kilograms, 20.2 kilograms, 20.4 kilograms, 21 kilograms, 21.5 kilograms, 22.6 kilograms, 22.7 kilograms, 22.8 kilograms, 23.2 kilograms, 23.4 kilograms, 24 kilograms, 24.9 kilograms, 26 kilograms, 26.1 kilograms, 26.7 kilograms, 28 kilograms, 28.4 kilograms, 30 kilograms, 31.5 kilograms, 32 kilograms, 32.1 kilograms, 33.5 kilograms, 34 kilograms, 34.4 kilograms.

Here is a histogram showing the same distribution.

In this case, it is difficult to make sense of the distribution from the dot plot because the precision of the measurement means the dots are distinct and so close together. The histogram of the same data set does a much better job showing the distribution of weights by grouping similar values to show an overall trend, even though we can’t see the individual data values.

The mean (also called the average) is one number we use to describe a whole data set. To find the mean, we use a 3-step procedure:

1. ADD all the values in the data set.
2. COUNT how many values there are. Call this number $n$.
3. DIVIDE the sum by $n$.

For example, to find the mean of $\{4, 6, 8, 6, 6\}$:

• Sum: $4 + 6 + 8 + 6 + 6 = 30$
• Count: there are $5$ values.
• Mean: $30 \div 5 = 6$.

So the mean of this data set is 6.

The procedure works the same way for decimal data sets. To find the mean of $\{2.5, 3.0, 3.5, 4.0, 2.0\}$: sum is $15.0$, count is $5$, mean is $15.0 \div 5 = 3.0$.

Sometimes a general description of a distribution does not give enough information, and a more precise way to talk about center or spread would be more useful. The mean, or average, is a number we can use for the center to summarize a distribution.

We can think about the mean in terms of “fair share” or “leveling out.” That is, a mean can be thought of as a number that each member of a group would have if all the data values were combined and distributed equally among the members.

For example, suppose there are 5 containers, each of which has a different amount of water: 1 liter, 4 liters, 2 liters, 3 liters, and 0 liters.

There are 5 identical tape diagrams that are each partitioned into 4 equal parts. The first diagram has 1 part shaded. The second diagram has 4 parts shaded. The third diagram has 2 parts shaded. The fourth diagram has 3 parts shaded. The fifth diagram has no parts shaded.

To find the mean, first we add up all of the values. We can think of this as putting all of the water together: $1+4+2+3+0=10$.

  

  

To find the “fair share,” we divide the 10 liters equally into the 5 containers: $10\div 5 = 2$.

The mean is useful when each unit of measurement has equal importance. For example, it may make sense to find the mean score of assignments of the same importance, such as all quizzes. If some grades are more important, it may not make sense to find the mean. For example, it may not make sense to find the mean score when there are 6 short homework assignments and one major essay.

Suppose the quiz scores of a student are 70, 90, 86, and 94. We can find the mean (or average) score by finding the sum of the scores $(70+90+86+94=340)$ and dividing the sum by four $(340 \div 4 = 85)$. We can then say that the student scored, on average, 85 points on the quizzes.

In general, to find the mean of a data set with $n$ values, we add all of the values and divide the sum by $n$.

The median is the middle value of a data set when the values are listed from least to greatest. Half of the values are at or below the median, and half are at or above the median.

To find the median:

1. Order the values from least to greatest.
2. Cross off from both ends of the list — one from the left, one from the right — until you reach the middle.
3. If there is one middle value (an odd number of values), that value is the median.
4. If there are two middle values (an even number of values), the median is the average of those two values: add them and divide by 2.

Odd N example: The data set 4, 7, 9, 12, 15 has 5 values. The middle (3rd) value is 9, so the median is 9.

Even N example: The data set 6, 10, 14, 18 has 4 values. The two middle values are 10 and 14. The median is (10 + 14) / 2 = 24 / 2 = 12.

Split-the-difference example: The data set 18, 22, 25, 27 has 4 values. The two middle values are 22 and 25. The median is (22 + 25) / 2 = 47 / 2 = 23.5. The median doesn't have to be a whole number, and it doesn't have to be a value that's actually in the data set.

The median is another measure of center for a distribution. It is the middle value in a data set when values are listed in order. The number of values less than or equal to the median is the same as the number of values that are greater than or equal to the median.

To find the median, we order the data values from least to greatest and find the number in the middle.

Suppose we have 5 dogs whose weights, in pounds, are shown in the table. The median weight for this group of dogs is 32 pounds because three dogs weigh less than or equal to 32 pounds and three dogs weigh greater than or equal to 32 pounds.

Now suppose we have 6 cats whose weights, in pounds, are listed here. Notice that there are 2 values in the middle: 7 and 8.

The median weight must be between 7 and 8 pounds, because half of the cats weigh less than or equal to 7 pounds, and half of the cats weigh greater than or equal to 8 pounds.

When there are even numbers of values, we take the number exactly in between the two middle values. In this case, the median cat weight is 7.5 pounds because $(7+8)\div 2=7.5$.

The mean and the median are two different ways to describe the center of a data set. Sometimes they're equal. Sometimes they're very different. Looking at both side-by-side tells us about the shape of the distribution.

To compute the mean: add all the values, then divide by the number of values.

To compute the median: order the values from least to greatest, then find the middle (or average the two middle values for even N).

Symmetric example: The data set 4, 6, 8, 6, 6 has mean (4 + 6 + 8 + 6 + 6) / 5 = 30 / 5 = 6. Sorted: 4, 6, 6, 6, 8. The median is the 3rd value, 6. Mean = median = 6.

Skewed-right example: The data set 2, 3, 4, 5, 21 has mean (2 + 3 + 4 + 5 + 21) / 5 = 35 / 5 = 7. Sorted: 2, 3, 4, 5, 21. The median is 4. The mean (7) is much greater than the median (4) because the value 21 pulls the mean up. The median is not affected the same way.

The pattern:

• When a data set is roughly symmetric (no extreme values), mean ≈ median.
• When a data set has a value that's much greater than the others, the mean is pulled up and is greater than the median.
• When a data set has a value that's much smaller than the others, the mean is pulled down and is less than the median.

Both the mean and the median are ways of measuring the center of a distribution. They tell us slightly different things, however.

The dot plot shows the number of stickers on 30 pages. The mean number of stickers is 21 (marked with a triangle). The median number of stickers is 20.5 (marked with a diamond).

<p>A dot plot for stickers on a page. The numbers 8 through 34, in increments of 2, are indicated. A diamond is indicated at 20.5 stickers and a triangle is indicated at 21 stickers. Data are as follows: 9 stickers, 1 dot; 10 stickers, 1 dot; 11 stickers, 2 dots; 12 stickers, 1 dot; 14 stickers, 1 dot; 16 stickers, 2 dots; 17 stickers, 1 dot; 18 stickers, 2 dots; 19 stickers, 1 dot; 20 stickers, 3 dots; 21 stickers, 1 dot; 22 stickers, 3 dots; 23 stickers, 1 dot; 24 stickers, 2 dots; 26 stickers, 2 dots; 28 stickers, 1 dot; 30 stickers, 1 dot; 32 stickers, 2 dots; 33 stickers, 1 dot; 34 stickers, 1 dot.</p>  

The mean tells us that if the number of stickers were distributed so that each page has the same number, then each page would have 21. We could also think of 21 stickers as a balance point for the number of stickers on all of the pages in the set. 

The median tells us that half of the pages have more than 20.5 stickers and half have less than 20.5 stickers. In this case, both the mean and the median could describe a typical number of stickers on a page because they are fairly close to each other and to most of the data points.

Here is a different set of 30 pages with stickers. It has the same mean as the first set, but the median is 23 stickers.

<p>A dot plot for “stickers on a page.” The numbers 8 through 34, in increments of 2, are indicated. A triangle is indicated at 21 stickers, and a diamond is indicated at 23 stickers. The data are as follows: 9 stickers, 1 dot; 10 stickers, 1 dot; 13 stickers, 1 dot; 14 stickers, 1 dot; 16 stickers, 1 dot; 17 stickers, 1 dot; 19 stickers, 1 dot; 20 stickers, 2 dots; 21 stickers, 2 dots; 22 stickers, 3 dots; 23 stickers, 6 dots; 24 stickers, 5 dots; 25 stickers, 4 dots; 26 stickers, 1 dot.</p>  

In this case, the median is closer to where most of the data points are clustered and is therefore a better measure of center for this distribution. That is, it is a better description of the typical number of stickers on a page. The mean number of stickers is influenced (in this case, pulled down) by a handful of pages with very few stickers, so it is farther away from most data points.

In general, when a distribution is symmetrical or approximately symmetrical, the mean and median values are close. But when a distribution is not roughly symmetrical, the two values tend to be farther apart.