Outliers

Outliers are data points that lie significantly outside the range of most other values in a dataset. Identifying outliers is important in statistics, as they can affect measures like the mean and standard deviation. There are two common methods to find outliers: using quartiles or using the mean and standard deviation.

1. Finding Outliers Using Quartiles:

Outliers can be identified using the interquartile range (IQR). The IQR is the range between the first quartile ($Q_1$) and the third quartile ($Q_3$).

Steps to Identify Outliers Using Quartiles:

1. Find $Q_1$ and $Q_3$: Arrange the data in ascending order and calculate the lower quartile ($Q_1$) and upper quartile ($Q_3$).
2. Calculate the IQR: Subtract $Q_1$ from $Q_3$: $$\text{IQR}=Q_3-Q_1$$
3. Determine the outlier thresholds: Use the following formulas to find the lower and upper thresholds:
   • Lower threshold: $Q_1-1.5\times \text{IQR}$
   • Upper threshold: $Q_3+1.5\times \text{IQR}$
4. Identify outliers: Any data point below the lower threshold or above the upper threshold is an outlier.

Example: A dataset contains the following values: $2,4,5,7,9,12,14,18,22$.

• Step 1: Find $Q_1$ and $Q_3$:
  • $Q_1=5$ (lower quartile)
  • $Q_3=14$ (upper quartile)
• Step 2: Calculate the IQR: $Q_3-Q_1=14-5=9$.
• Step 3: Find the thresholds:
  • Lower threshold: $5-1.5\times 9=-8.5$.
  • Upper threshold: $14+1.5\times 9=27.5$.
• Step 4: Identify outliers: Any data point below $-8.5$ or above $27.5$ is an outlier. In this case, there are no outliers.

2. Finding Outliers Using Mean and Standard Deviation:

Outliers can also be identified by comparing data points to the mean and standard deviation of the dataset. A common rule is that any data point more than 2 or 3 standard deviations away from the mean is considered an outlier.

Steps to Identify Outliers Using Mean and Standard Deviation:

1. Find the mean ($\mu$) and standard deviation ($\sigma$): Calculate the average and standard deviation of the dataset.
2. Determine the thresholds: Use the following formulas to find the lower and upper thresholds:
   • Lower threshold: $\mu -2\sigma$ (or $\mu -3\sigma$)
   • Upper threshold: $\mu +2\sigma$ (or $\mu +3\sigma$)
3. Identify outliers: Any data point outside the thresholds is an outlier.

Example: A dataset contains the following values: $10,12,15,18,20,25,50$.

• Step 1: Find the mean and standard deviation:
  • $\mu =\frac{10+12+15+18+20+25+50}{7}=21.43$.
  • $\sigma =13.27$ (calculated using the standard deviation formula).
• Step 2: Determine the thresholds (using $2\sigma$):
  • Lower threshold: $21.43-2\times 13.27=-5.11$.
  • Upper threshold: $21.43+2\times 13.27=47.97$.
• Step 3: Identify outliers: Any data point outside $-5.11$ and $47.97$ is an outlier. Here, $50$ is an outlier.

3. Summary:

• Outliers can be found using the IQR method or the mean and standard deviation method.
• The IQR method uses quartiles and thresholds of $1.5\times \text{IQR}$ to identify outliers.
• The mean and standard deviation method identifies data points outside $2\sigma$ or $3\sigma$ from the mean as outliers.