Here we talk through components involved in understanding what *dot plots* are, how to read them, and some aspects overall for how to interpret them. This is **not** a list of questions to ask every time your students work with *dot plots*, but rather a resource of skills regarding *dot plots* that we need to help our students gain over time.

*What are dot plots?*

A *dot plot* shows each data point (case) for an attribute along a numberline (one axis) to investigate the number or frequency of cases for each value. If desired, another attribute may be included within the formatting of the data points or along the other axis.

*How do we read dot plots?*

**What is the range of the quantitative scales/x-axis?**

**Tuva tip:** Besides just reading the lowest and highest values on the x-axis, students can adjust the size of the axis by moving the “T” bars at each end to see if that is the full extent of the data, and/or look at the data values in the corresponding attribute column in the Table View (below graph).

**What does each dot on the graph represent?**

**Tuva tip:** Have students open up the Table View (below graph), let them click on any row and see the corresponding case get highlighted in the graph. The details of the case can also be read from the case card which slides out on clicking a dot on the graph. Reinforce the connection between the graph, data table, and the case card.

**How many data values are in the graph?**

**Tuva tip:** Use the Count function or have students look through the number of cases in the Table View (below graph).

**Moving from left to right, do the values tend to increase or decrease?****On which end of the scale will you find the highest data values? Where will you find the smallest data values?**

**Tuva tip:** Encourage students to click on different dots or hover over them to read the data values of the dots.

**What value occurs the most frequently?**

**Tuva tip:** Ensure that kids are understanding that the number of dots vertically stacked above a specific x-axis value indicates the frequency by having them look at the data values in the corresponding attribute column in the Table View (below graph). For example, what life expectancy value occurs the most frequently? There are 5 dots (dog breeds) at 14 years, so it is the most frequent life expectancy value.

**Are there any extreme values or outliers in the data?**

**Tuva tip:** Do it by eye for beginners, encourage them to study the context of the data to see if it makes sense. For example, a difference of a few degrees in temperature at a location as compared to the rest of the temperature values may not necessarily qualify the data point as an outlier but a difference of the same magnitude inside of a refrigerator may be significant. Use dividers and percentages to estimate outliers as precursor to using IQR to find outliers.

**How many cases are above a certain value?**

**Tuva tip:** Use the dividers and the count function for larger distributions. For example, how many dots (dog breeds) have a life expectancy value greater than 12 years? There are 7 dots above 12 years (at 13, 14, or 16 years).

**What percentage of data values are above a certain value?**

**Tuva tip:** Use the dividers and percentages. This application of percentages enables a shift from additive to multiplicative thinking.

**What is the maximum value?**

**Tuva tip:** Always use the context of the data for framing all data interpretation related questions. For example, rather than asking “What is the maximum value?” ask questions like “What is the highest life expectancy in this group?” or “Which breed has the highest life expectancy?”

**What is the minimum value?**

**Tuva tip:** Similar to the maximum value questions, always use the context of the data for framing all data interpretation related questions.

**What is the frequency of cases for a given value?**

**Tuva tip:** Again, always use the context of the data for framing all data interpretation related questions. For example, the question “how many dog breeds have a life expectancy of 11 years?” helps students key into frequency of cases for a given value that is relevant for their interpretation of the data.

**If there are multiple series (sets of data), how do the maximum, minimum, and frequency of values compare within and among series?**

*How do we interpret distribution using dot plots?*

**What is the shape of the distribution?**

Tuva tip: Use informal vocabulary for beginners, use the pencil tool to trace out the overall shape, use the reference line to test for symmetry to compare the shape and number of dots on either side of the reference line.

**Are there any clumps/clusters or humps in the distribution?**

**Tuva tip:** Use the annotate tool to mark distinctive features and focus on where these are located on the axis.

**What is typical of this group?**

**Tuva tip:** Use dividers and percentages to mark the central clump, have the students enclose the middle 50% of the data. Visualizing the center as a range helps in appreciating variability. This also aids in an easy transition to working with box plots.

**What is the middle value?**

**Tuva tip:** Visually locate the median using dividers and the count function for a more thorough understanding of the measure rather than just calculating the median mathematically.

**How many data values are above the middle value?**

**Tuva tip:** Use dividers, Count and/or Percent to reinforce the understanding of the median.

**For different distributions shapes, how does the the order in which the mean and median appear vary?**

**Tuva tip:** Use multiple datasets to demonstrate different distributions or filter out tails of skewed distributions.

**How would skewness, extreme values, and outliers affect the mean or the median of the data?**

**Tuva tip:** Test by filtering out outliers and studying how the values of mean and the median change of the data with and without those data points.

**What happens to the difference between mean and median as you keep filtering out extreme values, or as the distribution becomes more and more symmetrical?****What quantitative measure of center would you use to represent what is typical for a group?**

**Tuva tip:** It is important to emphasize here that both measures have their uses. It’s just the characteristic of the mean that it is more sensitive to skewness, extreme values, or outliers. Apart from the shape, the purpose of the investigation plays a critical role in choosing the measure, too. For example, mean is super useful (and median is not) if you need to make calculations connecting individuals to the population. For example, the mean income of people in a country is $45,000. What is the total amount people earn? Answer: $45,000 x (population). You might want to know that to judge how much money is in the economy. But if you wanted to get an idea about how much people earn on an average--median is the measure from the data that will give you that answer.

**Are the data values tightly clustered around the center or spread out?**

**Tuva tip:** Use the pencil tool to trace out the overall shape, use the reference line at the center to compare the shape and number of dots on either side of the reference line.

**Does the center give you a fair idea of what is typical for the group?**

**Tuva tip:** Use dividers and percentages to mark the central clump, have the students enclose the middle 50% of the data. Visualizing the center as a range helps in appreciating variability. This also aids in an easy transition to working with box plots.

**How variable are the data? Should we just use the minimum and maximum values to describe the variability?****What are the advantages of enclosing the middle 50% of the values to describe the spread?****If you shrink the range below 50%, would you be describing the spread adequately?****How many data points (or percentage of data points) will you trim on either ends to get an idea about the spread?**

**Tuva tip:** Use dividers to trim off an equal number of points from either ends--traditionally 5-10%.

*How do we compare groups with dot plots?*

**How are the two dot plots similar? How are they different?****How do the means of the two distributions compare?****What is the difference in means?****How do the spreads (middle 50%) compare?****Is the middle 50% of one distribution wider/narrower than the other?****Is the middle 50% of one distribution higher than the other?****Do the middle 50% of the two distributions overlap? Is there a lot of overlap?****In case, the middle 50% of both distributions have similar ranges, ask: what is the range of the middle 50% of the data?**

**Tuva tip:** Use the range (spread units) to judge if the difference in means of the two distributions is big enough to be meaningful. For example, difference in means = 5 cm and range of the middle 50% = 10 cm. Divide diff.in means by range = 5/10 = 0.5. That is, 50% is quite big compared to the variability within each group.

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