Math Snap

STEP 1

What is this asking?
We need to find the unusual number in Mary's jogging data.
Watch out!
Don't just pick the biggest or smallest number; an outlier is a number that's significantly different from the others.

STEP 2

1. Arrange the Data
2. Calculate the Quartiles
3. Calculate the IQR
4. Identify the Outlier

STEP 3

Let's put the jogging distances in order from least to greatest!
This helps us see the spread of the data.
Our data is $0, 5, 6, 6, 6, 7, 8, 9, 12$.

STEP 4

Now, let's find the median (the middle value).
In our ordered list, the median is $6$.

STEP 5

Next, we'll find the first quartile (Q1), the median of the lower half of the data (excluding the overall median if the dataset has an odd number of elements).
The lower half is $0, 5, 6, 6$, so Q1 is the average of $5$ and $6$, which is $\frac{5 + 6}{2} = \frac{11}{2} = 5.5$.

STEP 6

Let's find the third quartile (Q3), the median of the upper half of the data (excluding the overall median if the dataset has an odd number of elements).
The upper half is $7, 8, 9, 12$, so Q3 is the average of $8$ and $9$, which is $\frac{8 + 9}{2} = \frac{17}{2} = 8.5$.

STEP 7

The interquartile range (IQR) is the difference between Q3 and Q1.
It tells us how spread out the middle $50\%$ of the data is.
So, IQR $= 8.5 - 5.5 = 3$.

STEP 8

An outlier is typically a value that falls below $Q1 - 1.5 \cdot \text{IQR}$ or above $Q3 + 1.5 \cdot \text{IQR}$.
Let's calculate these boundaries:

STEP 9

Lower boundary: $5.5 - 1.5 \cdot 3 = 5.5 - 4.5 = 1$.

STEP 10

Upper boundary: $8.5 + 1.5 \cdot 3 = 8.5 + 4.5 = 13$.

STEP 11

Any value outside these boundaries is considered an outlier.
Looking back at our data, only $0$ is less than $1$, so $0$ is the outlier!

The outlier of the data set is $0$.