Section 3.4

Measures of Position and Outliers

Learn how to standardize data with Z-scores, identify where values fall within a distribution using percentiles, and detect unusual observations (outliers).

1

The Z-Score (Standardized Values)

Definition

The Z-Score (or Standardized Value) represents the number of standard deviations that an observation is from the mean. It tells us how "unusual" a data point is.

Population Z-Score

= population mean, = population std dev

Sample Z-Score

= sample mean, = sample std dev

LogicLens: Why Z-Scores Are Powerful

  • • Dimensionless: Z-scores have no units, so you can compare values from completely different datasets (e.g., SAT scores vs. ACT scores).
  • • Positive Z: The value is above the mean.
  • • Negative Z: The value is below the mean.
  • • Z = 0: The value equals the mean exactly.

Example: Comparing Test Scores

Alice scored 1200 on the SAT (mean = 1060, std dev = 217).
Bob scored 28 on the ACT (mean = 21, std dev = 5.2).
Who performed better relative to their peers?

Alice's Z-Score

Bob's Z-Score

Bob's Z-score (1.35) is higher, so Bob performed better relative to his peers.

Z-Score Calculator

Quick Examples:

2

Percentiles & Quartiles

Percentiles

The -th Percentile () is the value below which percent of the data falls.

Example: If you score in the 90th percentile, you scored higher than 90% of all test-takers.

Quartiles

Quartiles divide the data into four equal parts:

  • • = 25th Percentile ()
  • • = 50th Percentile () = Median
  • • = 75th Percentile ()

The Interquartile Range (IQR)

The IQR measures the spread of the middle 50% of the data. It is resistant to outliers because it ignores the extreme values in the top and bottom quarters of the dataset.

Example: Finding Quartiles

Given the ordered dataset: 3, 5, 7, 8, 12, 15, 18, 21

QuartilePositionValue
Median of lower half: (5 + 7)/26
(Median)(8 + 12)/210
Median of upper half: (15 + 18)/216.5
3

Identifying Outliers (The 1.5 × IQR Rule)

What is an Outlier?

An outlier is an observation that is unusually far from the rest of the data. Outliers can indicate data entry errors, measurement issues, or genuinely unusual cases that warrant further investigation.

The Fence Method (Step-by-Step)

  1. 1Find (25th Percentile) and (75th Percentile)
  2. 2Calculate
  3. 3Calculate the Fences:

Lower Fence

Upper Fence

Any data point below the Lower Fence or above the Upper Fence is classified as an outlier.

Example: Outlier Detection

Given: , . Is the value 150 an outlier?

Since 150 is exactly at the Upper Fence, it is typically not considered an outlier (we look for values beyond the fences). However, values like 151 would be flagged as outliers.

Try It Yourself

Percentiles, Quartiles & Outlier Calculator

Employee Ages (No Outliers): Ages of 45 employees at a mid-sized company
(45 data points)
th percentile
Min
23.00
Max
55.00
Mean
37.27
Count
45

Quartiles

(25th)
31.00
(Median)
36.00
(75th)
43.00

IQR & Fences

IQR
12.00
Lower Fence
13.00
Upper Fence
61.00

No Outliers Detected

All 45 data points fall within the acceptable range [13.00, 61.00].

Sorted Data

232425262728282929303031313232333334343535363637373838393940414242434445464748495051525355
LogicLens Practice

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