Statistics - Z-Scores, Boxplots, Quartiles, Percentiles & Outliers

Z Scores

• Z-score indicates how many standard deviations a value is from its population mean.

• $z = 1$: value is one standard deviation above the mean.

• $z = -2$: value is two standard deviations below the mean.

Computing a Z Score

• Let $x$ be a value from a population with mean $\mu$ and standard deviation $\sigma$.
• The z-score for $x$ is calculated as: $z = \frac{x - \mu}{\sigma}$.

Example

• Mean height for adult men in the US: $\mu = 69.4$ inches, $\sigma = 3.1$ inches.

• Mean height for adult women in the US: $\mu = 63.8$ inches, $\sigma = 2.8$ inches.

• Man's height: 73 inches. Woman's height: 68 inches.

• Z-score for the man's height:

  • $z = \frac{73 - 69.4}{3.1} = 1.16$.

• Z-score for the woman's height:

  • $z = \frac{68 - 63.8}{2.8} = 1.5$.

• The woman is taller relative to the population because of a higher z score.

Empirical Rule and Z Scores

• For bell-shaped populations:
  • Approximately 68% of data have z scores between -1 and 1.
  • Approximately 95% of data have z scores between -2 and 2.
  • Almost all data have z scores between -3 and 3.

Boxplots

• Boxplot: A graph presenting the five-number summary and additional data information.
• Modified boxplot: a type of boxplot.

Constructing a Box Plot

• Data: Number of students absent in a middle school in Northwestern Montana during January.
• Step 1: Compute quartiles using technology (e.g., TI-84 Plus).
  • $Q_1 = 45$
  • $Q_2 \text{ (median)} = 51$
  • $Q_3 = 59$
• Step 2: Draw vertical lines at Q1, Q2, and Q3; complete the box with horizontal lines.
• Step 3: Calculate the interquartile range (IQR).
  • $IQR = Q<em>3 - Q</em>1 = 59 - 45 = 14$.
• Compute outlier boundaries:
  • Lower outlier boundary: $Q_1 - 1.5 \times IQR = 45 - 1.5 \times 14 = 24$.
  • Upper outlier boundary: $Q_3 + 1.5 \times IQR = 59 + 1.5 \times 14 = 80$.
• Step 4: Find the largest data value less than the upper boundary (77) and draw a horizontal line from $Q_3$ to it.
• Step 5: Find the smallest data value greater than the lower boundary (41) and draw a horizontal line from $Q_1$ to it.
• Step 6: Identify outliers (e.g., 100) and plot them separately.

Skewness and Boxplots

• Right Skew:
  • Median closer to $Q<em>1$ than $Q</em>3$.
  • Upper whisker longer than lower whisker.
• Left Skew:
  • Median closer to $Q<em>3$ than $Q</em>1$.
  • Lower whisker longer than upper whisker.
• Symmetric:
  • Median approximately halfway between $Q<em>1$ and $Q</em>3$.
  • Whiskers approximately equal in length.

Quartiles

• Quartiles divide a dataset into four equal parts.
• Every dataset has three quartiles: $Q<em>1$, $Q</em>2$, and $Q_3$.
  • $Q_1$: separates the lowest 25% from the highest 75%.
  • $Q_2$: (median) separates the lowest 50% from the highest 50%.
  • $Q_3$: separates the lowest 75% from the highest 25%.

Calculating Quartiles

1. Arrange data in increasing order.
2. Let $n$ = number of values.
   • For $Q_1$: $L = 0.25 \times n$.
   • For $Q_3$: $L = 0.75 \times n$.
3. If $L$ is a whole number, the quartile is the average of the values in positions $L$ and $L+1$.
4. If $L$ is not a whole number, round up to the next whole number, and the quartile is the value in that position.
5. $Q_2$ is the median.

Example

• Annual rainfall in Los Angeles during February over several years (45 values, already sorted).
  • For $Q<em>1$: $L = 0.25 \times 45 = 11.25 \approx 12$. $Q</em>1 = 0.92$.
  • For $Q<em>3$: $L = 0.75 \times 45 = 33.75 \approx 34$. $Q</em>3 = 4.89$.
  • Median: $Q_2 = 3.21$.

Five-Number Summary

• Consists of: minimum, $Q<em>1$, median, $Q</em>3$, maximum.
• Rainfall data summary: 0.14, 0.92, 3.21, 4.89, 13.68.

Using Technology for Quartiles

• Different technologies may use different procedures for finding quartiles.
• Example using TI-84 Plus calculator.

Detecting Outliers

• Outlier: A value much larger or smaller than the other values in a data set.
• Outliers can result from errors or reflect extreme values in the population.

IQR Method

1. Find $Q<em>1$ and $Q</em>3$.
2. Compute $IQR = Q<em>3 - Q</em>1$.
3. Compute outlier boundaries:
   • Lower boundary: $Q_1 - 1.5 \times IQR$.
   • Upper boundary: $Q_3 + 1.5 \times IQR$.
4. Any data value below the lower boundary or above the upper boundary is an outlier.

Example

• Absent students data: $Q<em>1 = 45$, $Q</em>3 = 59$.
  • $IQR = 59 - 45 = 14$.
  • Lower boundary: $45 - 1.5 \times 14 = 24$.
  • Upper boundary: $59 + 1.5 \times 14 = 80$.
  • The value 100 is greater than the upper boundary and is an outlier.

Interpreting Quartiles

• The median divides the dataset into two parts.
• Quartiles divide a dataset into four parts.
  • $Q_1$: separates the lowest 25% from the highest 75%.
  • $Q_2$: (median) separates the lowest 50% from the highest 50%.
  • $Q_3$: separates the lowest 75% from the highest 25%.

Examples

• Kayla's high B grade is more likely to be on the third quartile.
• Zorida (Q1), Phoebe (Q3), and Joanne (median); Zorida had the shortest average sleep duration.

Percentiles

• Percentiles divide a dataset into hundredths.
• The $p^{th}$ percentile separates the lowest $p$% of the data from the highest $100-p$%.
• Example: The 1st percentile separates the lowest 1% from the highest 99%.

Calculating Percentiles

1. Arrange data in increasing order.
2. Let $n$ = the number of values in the dataset.
3. For the $p^{th}$ percentile, calculate $L = \frac{p}{100} \times n$.
4. If $L$ is a whole number, the percentile is the average of the numbers in positions $L$ and $L+1$.
5. If $L$ is not a whole number, round it up to the next higher whole number, and the percentile is the number in this position.

Example

• Rainfall data in Los Angeles (45 values, already sorted); find the 60th percentile.
  • $L = \frac{60}{100} \times 45 = 27$.
  • 60th percentile = $\frac{3.58 + 3.71}{2} = 3.645$.

Finding Percentile for a Given Value

1. Arrange data in increasing order.
2. Let $x$ be the value whose percentile is to be computed.
3. Percentile $= 100 \times \frac{\text{number of values less than } x + 0.5}{\text{number of data values}}$.
4. Round the result to the nearest whole number.

Example

• In 1989, rainfall was 1.9 inches; what percentile does this correspond to?
  • 17 values are less than 1.9.
  • Percentile $= 100 \times \frac{17 + 0.5}{45} = 38.9 \approx 39$. The value 1.9 corresponds to the 39th percentile.