SAT-55: Descriptive Statistics — Mean and Median
Compute and compare the mean and median, work backward from a known mean, and know which is better with skewed data.
SAT-55: Descriptive Statistics — Mean and Median
Description: The mean (average) and the median (middle value) are the two main "center" measures on the SAT. You must compute each, reverse the mean to find a missing value, and know when one is more trustworthy than the other.
Mean (average)
mean = (sum of all values) ÷ (how many values)
The mean uses every number, so it is sensitive to extreme values. (Oʻzbekcha: oʻrtacha — barcha qiymatlar yigʻindisini ularning soniga boʻlamiz.)
Median (middle)
First sort the values from low to high, then take the middle one. If there is an even count, the median is the average of the two middle values. (Oʻzbekcha: median — saralangan qatorning oʻrtasidagi qiymat.)
Worked Example 1 — both measures
Find the mean and median of 4, 8, 6, 10, 7.
- Mean = (4 + 8 + 6 + 10 + 7) ÷ 5 = 35 ÷ 5 = 7.
- Sorted: 4, 6, 7, 8, 10 → middle value = 7 (median).
Worked Example 2 — even count median
Find the median of 3, 9, 5, 12.
- Sorted: 3, 5, 9, 12. Two middle values are 5 and 9.
- Median = (5 + 9) ÷ 2 = 7.
Worked Example 3 — work backward from the mean
The mean of 5 test scores is 82. Four of them are 78, 85, 90, and 80. Find the fifth.
- Total needed = mean × count = 82 × 5 = 410.
- Sum of the four known = 78 + 85 + 90 + 80 = 333.
- Fifth score = 410 − 333 = 77.
(Oʻzbekcha: oʻrtachadan teskari ishlash uchun avval umumiy yigʻindini topamiz: oʻrtacha × son.)
Mean vs median — which to trust
An outlier (an unusually large or small value) drags the mean toward it but barely moves the median. So for skewed data (like incomes, where a few huge values exist), the median is the better "typical" value.
Rule of thumb: symmetric data → mean ≈ median; strongly skewed data → trust the median. (Oʻzbekcha: kuchli nosimmetrik maʼlumotlarda medianga ishonish toʻgʻriroq.)
How adding or changing a value shifts the center
The SAT often changes one number and asks how the mean and median respond. Two facts make this easy:
- The mean reacts to every change — making any value bigger raises the mean, even a little. So if you add a value above the current mean, the mean goes up; below it, the mean goes down.
- The median only cares about position. Changing an extreme value to something even more extreme often leaves the median exactly where it was, because the middle position hasn't moved.
Knowing this, you can frequently answer "what happens to the mean/median?" questions without recomputing anything from scratch. (Oʻzbekcha: oʻrtacha har qanday oʻzgarishga taʼsirchan, median esa faqat oʻrtadagi oʻringa bogʻliq.)
Practice 1
The mean of 6 numbers is 10. If five of them sum to 52, what is the sixth?
Show answer
Total = 10 × 6 = 60. Sixth = 60 − 52 = 8.
Practice 2
A data set is 2, 3, 3, 4, 100. Which is more typical of the data, the mean or the median, and why?
Show answer
Mean = 112 ÷ 5 = 22.4; median = 3. The 100 is an outlier that inflates the mean, so the median (3) better represents a typical value.
Key words — Kalit soʻzlar
- Mean (average) — oʻrtacha qiymat
- Median — median (oʻrta qiymat)
- Sum — yigʻindi
- Sort / Order — saralash / tartiblash
- Outlier — chetga chiquvchi qiymat
- Skewed — nosimmetrik (qiyshiq)
- Symmetric — simmetrik
- Data set — maʼlumotlar toʻplami
- Typical value — odatiy qiymat
Summary
- Mean = sum ÷ count; reverse it with total = mean × count.
- Median = middle of the sorted list (average the two middles if even count).
- Outliers pull the mean but not the median; for skewed data, prefer the median.