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SSAT Middle Level: Quantitative • Data Analysis and Graphs

Compute and compare averages from data.

Averages summarize data sets, like finding the typical score in your favorite game.

Explore the lesson 4 min read
SSAT Middle Level illustration with a test notebook, quantitative function, and reading book
Inside this lesson 10 sections 5 questions 7 figures

SECTION 1

The Story Behind Averages

People have used averages for thousands of years. Long ago, Babylonians calculated them for astronomy. In the 1800s, scientists like Francis Galton made averages popular in everyday life. Today, you see averages in sports scores and test grades. They help make sense of lots of numbers fast.

2000 BCE

Babylonians use averages to predict planet paths, like tracking a soccer ball's average speed.

1662 • Early Statistics

John Graunt studies London deaths to find average lifespans.

1880s • Modern Term

Francis Galton coins 'average' for typical values in nature.

Today

Computers calculate averages instantly for games and apps.

Averages solve the problem of too much data. Imagine 20 test scores. Which one shows how the class did? Averages give you that quick picture.

SECTION 2

Core Types of Averages

There are three main averages: mean (add up and divide), median (middle value), and mode (most common). Each works best in different situations. Mean uses every number. Median ignores extremes. Mode shows favorites.

1

Mean

Total sum divided by count. Great for even data.
2

Median

Middle number in ordered list. Best against outliers.
3

Mode

Most frequent value. Perfect for categories.
⚽

Quick Analogy

Think of averages like game stats. Mean is total points divided by games, like your season average. Median skips one bad game. Mode is your go-to move that scores most.

SECTION 3

Seeing Averages on a Number Line

Seeing Averages on a Number Line Data set: 2, 3, 5, 5, 9 0 1 2 3 4 5 6 7 8 9 10 2 3 5, 5 9 Mean = 4.8 Median = 5 Mode = 5 How to find each average: Mean = (2 + 3 + 5 + 5 + 9) ÷ 5 = 24 ÷ 5 = 4.8 Median = middle value of {2, 3, 5, 5, 9} = 5 Mode = most frequent value = 5 💡 The mean (4.8) is pulled slightly left by the lower values 2 and 3
Data points plotted on a number line. The mean pulls toward the outlier at 9. Median and mode stay at 5.

This diagram shows how averages differ. The big 9 pulls the mean right. Median splits the middle. Mode repeats twice. You can see why we compare them! Great job spotting the patterns.

SECTION 4

Formulas for Averages

Each average has a simple formula. Start with the data list. Order it for median. Count repeats for mode. You can do these without a calculator.

MEAN

mean=(x1+x2+…+xn)÷nmean = (x₁ + x₂ + … + xₙ) ÷ n

x₁ to xₙ are data values. n is the count. Like (2+3+5+5+9) ÷ 5 = 4.8.

MEDIAN

middlevalue(orderedlist)middle value (ordered list)

Odd count: middle one. Even: average of two middles. Order: 2,3,5,5,9 → 5.

MODE

mostfrequentvaluemost frequent value

5 appears twice. No mode or bimodal if tie.

SECTION 5

Comparing in Different Data Sets

Comparing Averages in Different Data Sets How data shape affects the relationship between mean and median Symmetric Data Set { 1, 2, 3, 4, 5 } Frequency 1 2 3 1 2 3 4 5 Value Mean = 3.0 Median = 3.0 Equal! Right-Skewed Data Set { 1, 2, 3, 3, 10 } Frequency 1 2 3 1 2 3 ··· 10 outlier Value Mean = 3.8 Median = 3.0 Mean > Median Key Insight: An outlier pulls the mean toward it, but the median stays put — they no longer match!
Histograms compare symmetric (bell shape) and skewed data. In symmetric, all averages match. Skewed outlier changes mean but not median.

Skewed data tilts like a lopsided stack of blocks. Mean slides toward the heavy side. Median stays center. You're getting good at this!

SECTION 6

Step-by-Step Calculation

Quiz scores: 70, 85, 85, 90, 60. Find all three averages. Order first: 60, 70, 85, 85, 90. Easy steps ahead. You can do it!

Basketball Free Throws Example

  1. 1

    Step 1 — Mean

    Sum: 60 + 70 + 85 + 85 + 90 = 390. Divide by 5: 390 ÷ 5 = 78
    78
  2. 2

    Step 2 — Median

    Ordered list middle: third value = 85
    85
  3. 3

    Step 3 — Mode

    85 appears twice: most frequent = 85
    85
💡

Tip

Low 60 pulls mean down. Median and mode show stronger performance. Compare wisely!

SECTION 7

When to Use Each Average

AverageStrengthLimit
MeanUses all data.Outliers change it.
MedianIgnores extremes.Ignores most data.
ModeShows most common.May not exist.

Choose based on your data!

★

Key takeaway

Like picking a bat in baseball: mean for total power, median for steady hits, mode for your favorite swing. Match to the game!

SECTION 8

Beyond Basic Averages

Basic averages lead to cool tools like range or quartiles. Range is max minus min. It shows spread. SSAT might ask to compare with averages.

BasicAdvanced
Mean, median, modeRange, quartiles, weighted mean
Center onlyCenter + spread

These build your skills for harder problems. Practice now, shine later!

SECTION 9

Test Your Skills

Question 1: CONCEPTUAL

What does the median represent? (A) Most frequent (B) Middle value (C) Total sum (D) Biggest number (E) Average difference
Show answer
B is correct. Median is the middle value in an ordered list, like the center of a line of kids by height. A is mode, C relates to mean, D is max, E is range.

Question 2: BASIC CALCULATION

Find the mean of 4, 6, 8. (A) 4 (B) 6 (C) 8 (D) 18 (E) 5
Show answer
B is correct. Sum 4+6+8=18, divide by 3: 18÷3=6. A too low, C too high, D is sum, E wrong divide. Show steps every time!

Question 3: INTERMEDIATE

Scores: 10, 20, 20, 30, 40. Median? (A) 20 (B) 25 (C) 30 (D) 24 (E) 20 and 30
Show answer
A is correct. Ordered: 10,20,20,30,40. Middle (third) is 20. B averages middles (even list), C fourth, D mean, E bimodal mode. Order first!

Question 4: APPLIED

Game scores: 5,6,7,8,50. Mean=15.2, median=7. Best for typical score? (A) Mean (B) Median (C) Both (D) Neither (E) Mode
Show answer
B is correct. Outlier 50 pulls mean high. Median shows usual 5-8 range, like ignoring one crazy game. A skewed by outlier, C no, D wrong, E no mode. Compare always!

Question 5: CRITICAL THINKING

House prices: $100k, $110k, $120k, $130k, $1M. Which average for 'typical' home? (A) Mean (B) Median (C) Mode (D) Range (E) Min
Show answer
B is correct. Median $120k ignores mansion outlier. Mean ~$292k too high. C no mode, D spread $900k, E lowest. Real estate uses median for truth!

Awesome work! Review weak spots and try again. You're SSAT ready.

SUMMARY

Key Points to Remember

Summary

Master mean (sum ÷ count), median (middle), and mode (most common). Compare for best fit—outliers hurt mean most.

Visuals like number lines show why. Practice builds speed. You've got this for the SSAT!

Varsity Tutors • SSAT Middle Level • Compute and compare averages from data.