STAT/MATH 3379 Elementary Statistics: Chapter 3 - Averages and Variation

Z-Score

• Let x be a value from a population with mean \mu and standard deviation \sigma.

• The z-score (z) for x represents how many standard deviations x is from its population mean.

• Formula: z = \frac{x - \mu}{\sigma}.

• A positive z-score means the value is above the mean; a negative z-score means it's below the mean.

Percentiles

• Divide a data set into hundredths.

• For a number p between 1 and 99, the p-th percentile separates the lowest p\% of the data from the highest (100 - p)\% .

• A raw score does not necessarily correspond to the same percentile; percentile indicates relative position.

Quartiles

• Divide a data set into four approximately equal pieces.

• First Quartile (Q_1): Separates the lowest 25\% of data from the highest 75\%.

• Second Quartile (Q2): Separates the lowest 50\% of data from the highest 50\%. Q2 is the same as the median.

• Third Quartile (Q_3): Separates the lowest 75\% of data from the highest 25\%.

Interquartile Range (IQR)

• A measure of variation that gives the range of the middle portion (about half) of the data.

• Formula: IQR = Q3 - Q1.

Detecting Outliers

• Step 1: Find Q1 and Q3.

• Step 2: Compute IQR = Q3 - Q1.

• Step 3: Compute outlier boundaries:

  • Lower Outlier Boundary = Q_1 - 1.5(IQR)

  • Upper Outlier Boundary = Q_3 + 1.5(IQR)

• Step 4: Any data value less than the lower outlier boundary or greater than the upper outlier boundary is an outlier.

Box-and-Whisker Plot

• Highlights important features of a data set using a five-number summary:

  • Minimum value

  • First Quartile (Q_1)

  • Median (Q_2)

  • Third Quartile (Q_3)

  • Maximum value