SAT-56: Mode, Range, and Outliers

SAT-56: Mode, Range, and Outliers

Description: Beyond mean and median (SAT-55), the SAT uses three more simple but important ideas: the mode (most frequent value), the range (spread from low to high), and outliers (extreme values). Knowing how an outlier changes each statistic is a favorite test trap.

Mode — the most common value

The mode is the value that appears most often. A data set can have one mode, more than one (if several values tie), or no mode (if nothing repeats). (Oʻzbekcha: moda — eng koʻp takrorlangan qiymat.)

Range — the spread

range = maximum value − minimum value

The range measures how spread out the data is. A big range means the values are far apart. (Oʻzbekcha: amplituda (range) — eng katta va eng kichik qiymat orasidagi farq.)

Outliers — the troublemakers

An outlier is a value much larger or much smaller than the rest. Outliers strongly affect the mean and range, but have little effect on the median and mode. This contrast is exactly what the SAT likes to test.

Worked Example 1 — mode and range

Find the mode and range of 3, 7, 7, 7, 9, 12.

• Mode = 7 (appears three times).
• Range = 12 − 3 = 9.

Worked Example 2 — outlier's effect on the mean

Data: 10, 11, 12, 13. Now add the value 70. What happens to the mean and the median?

• Before: mean = 46 ÷ 4 = 11.5; median = (11 + 12)/2 = 11.5.
• After: mean = 116 ÷ 5 = 23.2 (jumped a lot); median = 12 (barely moved).
• Conclusion: the outlier 70 inflated the mean but hardly touched the median.

(Oʻzbekcha: chetga chiquvchi qiymat oʻrtachani keskin oshiradi, medianni esa deyarli oʻzgartirmaydi.)

Worked Example 3 — outlier's effect on the range

Using the same data, what does adding 70 do to the range?

• Before: range = 13 − 10 = 3. After: range = 70 − 10 = 60.
• The outlier exploded the range, because range depends on the extremes.

Summary trap to remember: outliers move mean and range a lot, but median and mode very little. (Oʻzbekcha: outlier — oʻrtacha va amplitudaga kuchli, median va modaga kam taʼsir qiladi.)

When is a value really an outlier?

On the SAT you can usually identify an outlier just by looking — it is a point clearly detached from the rest of the data. You do not need a formal rule, but it helps to ask two questions: is this value far from the cluster, and is there a real reason it might be a mistake or a special case (a typo, a different kind of subject)? If a data set has an outlier, mention its effect when you compare statistics, because that is almost always the point of the question. Removing an outlier pulls the mean back toward the cluster and shrinks the range, while the median and mode barely budge. (Oʻzbekcha: outlier — qolgan maʼlumotlardan aniq ajralib turadigan qiymat; uni olib tashlasangiz, oʻrtacha va amplituda kichrayadi.)

Practice 1

Find the mode and range of 5, 5, 8, 9, 9, 9, 20.

Practice 2

A set is 4, 4, 5, 6, 6. If you change the 6's max to 100, which changes more — the median or the range?

Key words — Kalit soʻzlar

• Mode — moda
• Range — amplituda (tarqoqlik)
• Maximum / Minimum — eng katta / eng kichik
• Outlier — chetga chiquvchi qiymat
• Frequency — chastota (takrorlanish soni)
• Spread — tarqoqlik
• Affect — taʼsir qilish
• Extreme value — keskin (chetki) qiymat
• Data set — maʼlumotlar toʻplami

Summary

• Mode = most frequent value; range = max − min.
• Outliers strongly affect the mean and range.
• Outliers barely affect the median and mode.