Study guide: Algebra1: Descriptive Data and Outliers
Data summaries describe center and spread using stated definitions. Mean, median, interquartile range and mean absolute deviation respond differently to extreme values, so the chosen measure should match the question and convention.
Ideas to explain in your own words
- A mean divides the total by the number of values. A median locates the middle of ordered data; a high outlier can pull the mean upward while leaving the middle position unchanged. Sort before finding the median and count every value in the mean denominator.
- Interquartile range subtracts the lower quartile from the upper quartile. With the stated median-of-halves convention, each quartile is the middle value of its corresponding sorted half. Quartile conventions can differ across software and sample sizes.
- Mean absolute deviation averages each value’s nonnegative distance from the mean. Taking absolute values prevents positive and negative deviations from canceling, creating a genuine measure of spread. Use the mean of the data before computing distances.
- Replacing one endpoint of a sorted data set can increase the total without changing the middle value. The mean changes by the total change divided by the observation count. An unchanged median does not mean the data did not change.
Useful vocabulary
- mean
- Sum of values divided by their count.
- median
- The middle ordered value or average of two middle values.
- interquartile range
- Upper quartile minus lower quartile.
- mean absolute deviation
- Average absolute distance from the mean.
Watch your reasoning
- Sort before finding the median and count every value in the mean denominator. These complete supplied data need no population inference; report both summaries without pretending they are interchangeable.
- Quartile conventions can differ across software and sample sizes. Use the explicit convention given here; the IQR describes middle-half spread rather than the distance between the minimum and maximum.
- Use the mean of the data before computing distances. Average the absolute deviations using the full number of observations; signed deviations alone would sum to zero and hide variation.
A worked review example
Complete supplied guide examples: Compare an Arithmetic Mean with a Median — For the complete supplied data [2, 3, 4, 5, 16], which pair is (mean, median)? Candidate answers: (A) (16, 4) (B) (4, 6) (C) (6, 4) (D) (6, 5). The sum is 30; divide by 5 to get mean 6. The third sorted value is the median 4. The high value pulls the mean upward. Apply a Stated Quartile Convention — For sorted data [3, 5, 7, 9, 11, 13], use the median of each three-value half for Q1 and Q3. Find IQR. Candidate answers: (A) 10 (B) 2 (C) 4 (D) 6. Q1 is the lower-half middle value 5; Q3 is the upper-half middle 11. Thus IQR=11−5=6, under the stated convention. Average Absolute Distances from the Mean — For data [2, 4, 6], find mean absolute deviation from the mean, exactly. Candidate answers: (A) 0 (B) 2 (C) 4 (D) 4/3. The mean is 4. Absolute distances are 2,0,2. Their average is (2+0+2)/3=4/3; signed distances would incorrectly cancel. Separate Changes in Mean and Median — Data 2,3,4,5,6 change only by replacing the largest value with 21. Which statement is true? Candidate answers: (A) Both mean and median stay the same (B) The mean decreases (C) The mean increases; the median stays the same (D) The median increases by 15. The sum increases by 15, so the mean increases by 15/5=3. The middle sorted value remains 4. The outlier affects these centers differently.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.