Measures of Position in Statistics: Quartiles, Interquartile Range, Percentiles, and Standard Scores

Section 2.5: Measures of Position

Introduction

Measures of position are statistical tools used to describe the relative standing of data values within a data set. They help in interpreting and comparing individual scores, identifying outliers, and summarizing data distributions. Common measures include quartiles, percentiles, and standard scores (z-scores).

Quartiles

Definition and Properties

• Quartiles are values that partition an ordered data set into four equal parts.

• Each quartile contains approximately 25% of the data.

• First quartile (Q1): About one quarter of the data fall at or below this value.

• Second quartile (Q2): The median; about half the data fall at or below this value.

• Third quartile (Q3): About three quarters of the data fall at or below this value.

Example: Finding Quartiles

• Given a data set, order the values from least to greatest.

• Divide the data into four equal parts to find Q1, Q2, and Q3.

• Application: Quartiles are used to summarize data and identify the spread and center.

Using Technology to Find Quartiles

• Statistical software and calculators can compute quartiles efficiently.

• Input the data and use built-in functions to obtain Q1, Q2, and Q3.

Interquartile Range (IQR)

Definition and Calculation

• Interquartile Range (IQR) measures the spread of the middle 50% of the data.

• Calculated as the difference between the third and first quartiles:

$IQR = Q_3 - Q_1$

Using IQR to Identify Outliers

• Find Q1 and Q3.

• Multiply IQR by 1.5.

• Any data value less than $Q_1 - 1.5 imes IQR$ or greater than $Q_3 + 1.5 imes IQR$ is considered an outlier.

Example: Finding the Interquartile Range

• Given Q1 = 20 and Q3 = 35, the IQR is:

$IQR = 35 - 20 = 15$

• To identify outliers, calculate $Q_1 - 1.5 imes IQR$ and $Q_3 + 1.5 imes IQR$.

Box-and-Whisker Plot

Definition and Features

• A box-and-whisker plot is a graphical summary of a data set.

• Displays the minimum, Q1, median (Q2), Q3, and maximum values.

• Highlights the spread and center of the data, as well as potential outliers.

Steps to Draw a Box-and-Whisker Plot

1. Find the five-number summary: minimum, Q1, median, Q3, maximum.

2. Draw a box from Q1 to Q3 with a line at the median.

3. Draw whiskers from the box to the minimum and maximum values.

Example: Drawing a Box-and-Whisker Plot

• Given data: Min = 11, Q1 = 23, Q2 = 29, Q3 = 35, Max = 55.

• The box represents the middle 50% of the data, and whiskers extend to the minimum and maximum.

Percentiles and Other Fractiles

Definitions

• Percentiles divide a data set into 100 equal parts.

• Deciles divide a data set into 10 equal parts.

• Quartiles divide a data set into 4 equal parts.

Fractile	Divides Data Into	Symbol
Quartiles	4 equal parts	Q1, Q2, Q3
Deciles	10 equal parts	D1, D2, ..., D9
Percentiles	100 equal parts	P1, P2, ..., P99

Interpreting Percentiles

• The nth percentile is the value below which n% of the data fall.

• Percentiles are used to compare individual scores to the rest of the data set.

Finding the Percentile Corresponding to a Data Entry

• Use the formula:

$\text{Percentile of value} = \frac{\text{number of values less than data entry}}{\text{total number of values}} \times 100$

• Round to the nearest whole number.

Example: Finding Percentiles

• Given a data set, count the number of values less than the entry of interest.

• Apply the formula to determine the percentile rank.

Standard Score (z-score)

Definition and Calculation

• A standard score (z-score) measures the number of standard deviations a data value is from the mean.

• Calculated using the formula:

$z = \frac{\text{value} - \text{mean}}{\text{standard deviation}}$

Interpreting z-Scores

• A z-score of 0 indicates the value is equal to the mean.

• Positive z-scores indicate values above the mean; negative z-scores indicate values below the mean.

• z-scores allow comparison of values from different data sets with different means and standard deviations.

Example: Finding z-Scores

• Given a mean speed of 56 miles per hour and a standard deviation of 6 miles per hour, a speed of 72 miles per hour has a z-score:

$z = \frac{72 - 56}{6} = 2.7$

• This value is 2.7 standard deviations above the mean.

Comparing z-Scores from Different Data Sets

• z-scores standardize values, making it possible to compare scores from different distributions.

• For example, comparing heights of men and women using their respective means and standard deviations.

Summary Table: Measures of Position

Measure	Definition	Formula	Application
Quartile	Divides data into four equal parts	Q1, Q2, Q3	Summarize data, identify spread
Interquartile Range	Range of middle 50% of data	$IQR = Q_3 - Q_1$	Detect outliers, measure spread
Percentile	Value below which a given % of data fall	$\frac{\text{number less than entry}}{\text{total}} \times 100$	Rank, compare scores
z-score	Standardized score	$z = \frac{\text{value} - \text{mean}}{\text{standard deviation}}$	Compare across data sets

Conclusion

Measures of position are essential for understanding the distribution and relative standing of data values. They provide tools for summarizing data, identifying outliers, and making meaningful comparisons across different data sets.