SAT-64: Comparing Data Sets — Boxplots and Histograms

SAT-64: Comparing Data Sets — Boxplots and Histograms

Description: Two graphs summarize a whole data set at a glance: the boxplot shows its five key numbers and spread, and the histogram shows its shape. The SAT asks you to read each and to compare two data sets using center and spread.

The boxplot (box-and-whisker)

A boxplot is built from the five-number summary:

• Minimum and Maximum — the ends of the whiskers.
• Q1, Median (Q2), Q3 — the quartiles that form the box.
• IQR = Q3 − Q1, the interquartile range, is the width of the box (the middle 50% of the data).

The line inside the box is the median; a longer box or whisker means more spread on that side. (Oʻzbekcha: quti-diagramma besh asosiy sonni koʻrsatadi: min, Q1, median, Q3, max.)

The histogram (shape)

A histogram groups data into intervals and draws a bar for how many values fall in each. Its shape tells the story:

• Symmetric: roughly a mirror image; mean ≈ median.
• Skewed right: a long tail of large values pulls the mean above the median.
• Skewed left: a long tail of small values pulls the mean below the median.

(Oʻzbekcha: gistogramma maʼlumotlarning shaklini koʻrsatadi — simmetrik yoki bir tomonga choʻzilgan.)

Worked Example 1 — read a boxplot

A boxplot shows min 20, Q1 35, median 50, Q3 65, max 90. Find the IQR and the range.

• IQR = Q3 − Q1 = 65 − 35 = 30.
• Range = max − min = 90 − 20 = 70.

Worked Example 2 — interpret quartiles

Using that boxplot, about what percent of the data is below 35, and what percent is between 35 and 65?

• Q1 = 35 marks the bottom 25%, so about 25% is below 35.
• From Q1 to Q3 is the middle 50% of the data (between 35 and 65).

(Oʻzbekcha: Q1 — pastki 25%, Q1 dan Q3 gacha — oʻrtadagi 50%.)

Worked Example 3 — compare with a histogram's shape

A histogram of house prices has a long right tail. Is the mean or median larger, and which is more typical?

• Right-skewed → the mean is pulled up by the expensive houses, so mean > median.
• The median is the more typical price.

To compare two data sets, mention both center (median/mean) and spread (IQR/range). "Set A has a higher median and a smaller IQR" is a complete comparison. (Oʻzbekcha: ikki toʻplamni taqqoslashda markazni ham, tarqoqlikni ham aytib oʻting.)

Why boxplots and histograms show different things

These two graphs are not interchangeable. A histogram reveals the exact shape — whether the data is symmetric, skewed, or even has two peaks — because you can see every interval's height. A boxplot hides the shape but cleanly shows the quartiles and spread, which makes it ideal for comparing several data sets side by side. So if a question asks about shape or peaks, look to the histogram; if it asks about medians, quartiles, or which set is more spread out, look to the boxplot. Knowing which tool answers which question saves time. (Oʻzbekcha: gistogramma shaklni, quti-diagramma esa kvartillar va tarqoqlikni yaxshi koʻrsatadi.)

Practice 1

A boxplot has Q1 = 12 and Q3 = 28. What is the IQR, and what does it represent?

Practice 2

Histogram X is symmetric; histogram Y is skewed left. In which is the mean clearly below the median?

Key words — Kalit soʻzlar

• Boxplot — quti-diagramma
• Histogram — gistogramma
• Five-number summary — besh sonli xulosa
• Quartile (Q1, Q3) — kvartil
• IQR (interquartile range) — kvartillararo oraliq
• Skewed — bir tomonga choʻzilgan
• Symmetric — simmetrik
• Spread — tarqoqlik
• Center — markaz (oʻrta)

Summary

• Boxplot shows the five-number summary; IQR = Q3 − Q1 = middle 50%.
• Histogram shows shape: symmetric (mean ≈ median) or skewed (mean pulled toward the tail).
• Compare data sets with both center and spread.