2.3 Interpreting data

Achievable Praxis Core: Math (5733)

2. Data analysis, statistics, and probability

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Interpreting data

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Spotting patterns, trends, and outliers

A box-and-whisker plot (or boxplot) summarizes a data set using five key numbers: minimum, first quartile $Q_{1}$ , median $Q_{2}$ , third quartile $Q_{3}$ , and maximum. It shows the center and spread of the distribution.

Outliers are values that lie far from the rest of the data. The $IQR$ (interquartile range) is $Q_{3} - Q_{1}$ . The $1.5 \times IQR$ rule is the standard method for deciding whether a value qualifies. A value $x$ is an outlier if:

$Outlier if: x > Q_{3} + 1.5 \times IQR or x < Q_{1} - 1.5 \times IQR$

Values beyond this range are often plotted individually as points or small circles.
When outliers are present, the whiskers extend to the most extreme non-outlier values (the adjacent values inside the fence), not to the actual data minimum or maximum.
Outliers can indicate unusual cases, errors in data collection, or natural variability.

Definitions

Cluster: A concentration of data values within a specific range, indicating a subgroup or phase in the data. Clusters show where values tend to group together and can highlight common behaviors or conditions.
Outlier: A data value that is substantially higher or lower than the rest of the data set.
Trend: A general direction in which data values are moving over time or across categories. Trends can be increasing, decreasing, or constant, and they help reveal long-term patterns in a data set.

Patterns and trends describe how data values behave as a group - for example, steadily rising, steadily falling, or clustering around certain values. Outliers matter because they can distort summary measures like the mean or the range.

Finding the five-number summary

To find $Q_{1}$ and $Q_{3}$ : order the data, locate the median, then split into a lower half and an upper half. If $n$ is odd, exclude the median from both halves; if $n$ is even, split evenly. $Q_{1}$ is the median of the lower half and $Q_{3}$ is the median of the upper half.

We order the data, then find the median. We split the data into a lower half and an upper half to find the first and third quartiles. This gives the five-number summary for the data set.

Take the following data set:

$65, 68, 71, 74, 77, 82, 82, 89$

There are $n = 8$ values, already ordered. The median is the average of the $4^{th}$ and $5^{th}$ values:

$Q_{2} = \frac{74 + 77}{2} = 75.5$

The lower half is $65, 68, 71, 74$ , so:

$Q_{1} = \frac{68 + 71}{2} = 69.5$

The upper half is $77, 82, 82, 89$ , so:

$Q_{3} = \frac{82 + 82}{2} = 82$

This gives the five-number summary:

Minimum (Min) = $65$
First quartile ( $Q_{1}$ ) = $69.5$
Median ( $Q_{2}$ ) = $75.5$
Third quartile ( $Q_{3}$ ) = $82$
Maximum (Max) = $89$

Sidenote

Watch out

Always order the data before computing any quartile. A common mistake is reading $Q_{1}$ as the smallest value in the lower half - it’s actually the median of the lower half.

Visual interpretation

Some problems ask you to choose the correct box-and-whisker plot (also called a boxplot) for a given data set. To do that, you need to know how the five-number summary maps onto the picture.

The box spans from $Q_{1}$ to $Q_{3}$ , the median $Q_{2}$ is marked by a line inside the box, and whiskers extend from the box out to the minimum and maximum. Boxplots can be drawn horizontally or vertically; the box still spans from $Q_{1}$ to $Q_{3}$ with the median line inside - only the axis orientation changes.

Here’s what the five-number summary from the previous example looks like as a boxplot. Since no values fall outside the $1.5 \times IQR$ fences, the whiskers extend all the way to the actual minimum and maximum.

This example finds an outlier in a data set using the 1.5 IQR rule. It then compares the mean of all values to the mean without the outlier.

Example: Identify the outlier and compare means

$10, 12, 15, 14, 13, 100, 16, 14$

Find the following:

The outlier

The mean of all eight values

The mean of the seven typical values excluding the outlier

(spoiler)

Order the data: $10, 12, 13, 14, 14, 15, 16, 100$

First, compute $Q_{1}$ and $Q_{3}$ to find the IQR.

$Q_{1} = \frac{12 + 13}{2} = 12.5$
$Q_{3} = \frac{15 + 16}{2} = 15.5$
$IQR = Q_{3} - Q_{1} = 15.5 - 12.5 = 3$

Now apply the $1.5 \times IQR$ rule. A value is an outlier if it exceeds $Q_{3} + 1.5 \times IQR$ :

Upper outlier bound: $Q_{3} + 1.5 \times IQR = 15.5 + (1.5 \times 3) = 15.5 + 4.5 = 20$

Since $100 > 20$ , it is an outlier.

Mean of all values: $\frac{10 + 12 + 13 + 14 + 14 + 15 + 16 + 100}{8} = \frac{194}{8} = 24.25$
Mean without outlier: $\frac{94}{7} \approx 13.43$

Answer: The outlier is $100$ . The mean of all values is $24.25$ , and the mean without the outlier is approximately $13.43$ .

This example looks for clusters, trends, and outliers in an ordered data set.

Example: Detect trend, clusters, and outliers

Given the following data set, describe any clusters, trends, or outliers.

$5, 7, 8, 9, 10, 15, 16, 17, 20, 21$

(spoiler)

The data is already ordered from least to greatest.

Clusters: There are two visible clusters - one from $5$ to $10$ and another from $15$ to $21$ , with a gap in between.

Outliers: All values fall between $5$ and $21$ with no extreme gaps from the group, so there are no outliers.

Answer: The data show two clusters ( $5$ - $10$ and $15$ - $21$ ) and no outliers.

Justifying conclusions with data

Strong conclusions point to specific numbers, trends, or features in the display, and they avoid claims the data can’t support. When describing a trend, name the direction and support it with at least one specific value - for example, “visits increased by $300$ each month from January ( $1, 200$ ) to March ( $1, 800$ ).”

Correlation is not causation. A trend or association between two variables tells you they move together - it doesn’t tell you that one causes the other. Describing a trend accurately is fine; claiming a cause-and-effect relationship requires more than a pattern in the data.

Sidenote

Choosing the right display

Use the right display for your data type - this is a frequently tested skill.

Boxplot - numerical data; best for showing distribution and spotting outliers (e.g., comparing test score spreads across two classes)
Histogram - numerical data; best when the shape of the distribution matters. A symmetric histogram has roughly equal tails on both sides; a right-skewed histogram has a longer tail on the right (a few high values pull the mean up); a left-skewed histogram has a longer tail on the left.
Bar graph - categorical data; compares counts or values across distinct groups (e.g., favorite subject by number of students)
Line graph - data that changes over time; connects points to show trends (e.g., monthly rainfall over a year)
Scatterplot - two numerical variables; reveals relationships or correlations (e.g., hours studied vs. exam score)

Look for overall patterns, trends, clusters, or cycles before focusing on individual values
Identify any values that stand apart and confirm them as outliers using the $1.5 \times IQR$ rule
Use the five-number summary to describe the spread and center of a data set, and represent it visually with a boxplot
Anchor conclusions in specific numeric changes or clear features from the display
Avoid claims that extend beyond what the data actually show

Interpreting data

Spotting patterns, trends, and outliers

$Outlier if: x > Q_{3} + 1.5 \times IQR or x < Q_{1} - 1.5 \times IQR$

Values beyond this range are often plotted individually as points or small circles.
When outliers are present, the whiskers extend to the most extreme non-outlier values (the adjacent values inside the fence), not to the actual data minimum or maximum.
Outliers can indicate unusual cases, errors in data collection, or natural variability.

Definitions

Cluster: A concentration of data values within a specific range, indicating a subgroup or phase in the data. Clusters show where values tend to group together and can highlight common behaviors or conditions.
Outlier: A data value that is substantially higher or lower than the rest of the data set.
Trend: A general direction in which data values are moving over time or across categories. Trends can be increasing, decreasing, or constant, and they help reveal long-term patterns in a data set.

Finding the five-number summary

We order the data, then find the median. We split the data into a lower half and an upper half to find the first and third quartiles. This gives the five-number summary for the data set.

Take the following data set:

$65, 68, 71, 74, 77, 82, 82, 89$

There are $n = 8$ values, already ordered. The median is the average of the $4^{th}$ and $5^{th}$ values:

$Q_{2} = \frac{74 + 77}{2} = 75.5$

The lower half is $65, 68, 71, 74$ , so:

$Q_{1} = \frac{68 + 71}{2} = 69.5$

The upper half is $77, 82, 82, 89$ , so:

$Q_{3} = \frac{82 + 82}{2} = 82$

This gives the five-number summary:

Minimum (Min) = $65$
First quartile ( $Q_{1}$ ) = $69.5$
Median ( $Q_{2}$ ) = $75.5$
Third quartile ( $Q_{3}$ ) = $82$
Maximum (Max) = $89$

Sidenote

Watch out

Always order the data before computing any quartile. A common mistake is reading $Q_{1}$ as the smallest value in the lower half - it’s actually the median of the lower half.

Visual interpretation

Some problems ask you to choose the correct box-and-whisker plot (also called a boxplot) for a given data set. To do that, you need to know how the five-number summary maps onto the picture.

This example finds an outlier in a data set using the 1.5 IQR rule. It then compares the mean of all values to the mean without the outlier.

Example: Identify the outlier and compare means

$10, 12, 15, 14, 13, 100, 16, 14$

Find the following:

The outlier

The mean of all eight values

The mean of the seven typical values excluding the outlier

(spoiler)

Order the data: $10, 12, 13, 14, 14, 15, 16, 100$

First, compute $Q_{1}$ and $Q_{3}$ to find the IQR.

$Q_{1} = \frac{12 + 13}{2} = 12.5$
$Q_{3} = \frac{15 + 16}{2} = 15.5$
$IQR = Q_{3} - Q_{1} = 15.5 - 12.5 = 3$

Now apply the $1.5 \times IQR$ rule. A value is an outlier if it exceeds $Q_{3} + 1.5 \times IQR$ :

Upper outlier bound: $Q_{3} + 1.5 \times IQR = 15.5 + (1.5 \times 3) = 15.5 + 4.5 = 20$

Since $100 > 20$ , it is an outlier.

Mean of all values: $\frac{10 + 12 + 13 + 14 + 14 + 15 + 16 + 100}{8} = \frac{194}{8} = 24.25$
Mean without outlier: $\frac{94}{7} \approx 13.43$

Answer: The outlier is $100$ . The mean of all values is $24.25$ , and the mean without the outlier is approximately $13.43$ .

This example looks for clusters, trends, and outliers in an ordered data set.

Example: Detect trend, clusters, and outliers

Given the following data set, describe any clusters, trends, or outliers.

$5, 7, 8, 9, 10, 15, 16, 17, 20, 21$

(spoiler)

The data is already ordered from least to greatest.

Clusters: There are two visible clusters - one from $5$ to $10$ and another from $15$ to $21$ , with a gap in between.

Outliers: All values fall between $5$ and $21$ with no extreme gaps from the group, so there are no outliers.

Answer: The data show two clusters ( $5$ - $10$ and $15$ - $21$ ) and no outliers.

Justifying conclusions with data

Sidenote

Choosing the right display

Use the right display for your data type - this is a frequently tested skill.

Boxplot - numerical data; best for showing distribution and spotting outliers (e.g., comparing test score spreads across two classes)
Histogram - numerical data; best when the shape of the distribution matters. A symmetric histogram has roughly equal tails on both sides; a right-skewed histogram has a longer tail on the right (a few high values pull the mean up); a left-skewed histogram has a longer tail on the left.
Bar graph - categorical data; compares counts or values across distinct groups (e.g., favorite subject by number of students)
Line graph - data that changes over time; connects points to show trends (e.g., monthly rainfall over a year)
Scatterplot - two numerical variables; reveals relationships or correlations (e.g., hours studied vs. exam score)

Achievable Praxis Core: Math (5733)

2. Data analysis, statistics, and probability

Our Praxis Core: Math course is now in "early access" - get 50% off for a limited time.