Comparing Data Sets — Compare data sets using measures of center. - SHSAT Math

SECTION 1

Historical Context & Motivation

Imagine two basketball teams both claim they score "a lot" of points per game. How would you figure out which team really scores more? You need a single number that sums up each team's performance. That's exactly the problem mathematicians and scientists have worked on for centuries.

The idea of finding a measure of center (a single number that represents a whole data set) goes back hundreds of years. Early astronomers needed a way to combine many measurements of a star's position into one best guess. Over time, people developed the tools we now call the mean, median, and mode.

1500s • Early Averages

Astronomers like Tycho Brahe began averaging repeated observations of star positions to reduce errors in their data.

1700s • The Arithmetic Mean

Mathematicians formally defined the arithmetic mean (add up all values and divide by the count) as the standard way to find a center.

1800s • Median Gains Importance

Statisticians realized that the median (the middle value) was better when data had extreme values, like income data with a few very rich people.

Today • Comparing Data Sets

On tests like the SHSAT, you compare two or more data sets by looking at their measures of center side by side to draw conclusions.

The big question this lesson answers is: When you have two groups of numbers, how do you use the mean, median, and mode to compare them fairly?

SECTION 2

Core Principles & Definitions

Before you compare data sets, you need to understand the three main measures of center. Each one gives you a different way to describe the "middle" of a group of numbers.

Mean (Average)

Add up all the values in the data set, then divide by how many values there are. The mean uses every single number, so one very large or very small value can pull it up or down.

Median (Middle Value)

Put the numbers in order from least to greatest and find the one in the exact middle. If there are two middle numbers, average them. The median is not affected much by extreme values.

Mode (Most Frequent)

The value that appears the most often. A data set can have one mode, more than one mode, or no mode at all if every value appears the same number of times.

Comparing Two Data Sets

Calculate the same measure (mean, median, or mode) for both data sets, then compare the two numbers. The data set with the higher mean, for example, has a higher center on average.

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KEY TAKEAWAY

Think of each measure of center like a different camera angle at a soccer game. The mean is the wide-angle shot that captures everything (including that one crazy goal). The median is the center-field camera that ignores the edges. The mode is the instant-replay that shows the most common play. Each angle tells you something real, but you might trust one more than another depending on the situation.

SECTION 3

Visual Explanation

Let's look at two data sets side by side. Suppose Team A scored 10, 12, 14, 15, and 19 points in five games, while Team B scored 8, 11, 14, 16, and 21 points. The diagram below shows each team's scores on a number line, along with where their mean and median fall.

The cyan dots show Team A's scores and the violet dots show Team B's scores. The pink dashed lines mark each team's mean, and the gold dashed lines mark the median. Notice that even though the mean and median are the same for both teams, Team B's scores are more spread out.

This diagram shows a very important idea: two data sets can share the same mean and median but still look very different. That's why on the SHSAT, you might be asked to look at more than one measure or consider how spread out the data is before making a comparison.

SECTION 4

Mathematical Framework

Let's write down the formulas you need. These are simple, but getting them right is the key to SHSAT data-analysis questions.

MEAN (AVERAGE)

Mean = (Sum of all values) ÷ (Number of values)

Add up every number in the data set. Then divide by the total count. For example, if a data set is {3, 7, 10}, the mean = (3 + 7 + 10) ÷ 3 = 20 ÷ 3 ≈ 6.67.

MEDIAN (MIDDLE VALUE)

Odd count → middle value | Even count → (middle₁ + middle₂) ÷ 2

First, sort the numbers from smallest to largest. If there is an odd number of values, the median is the one right in the center. If there is an even number, take the two center values, add them, and divide by 2.

MODE (MOST FREQUENT)

Mode = value that appears most often

Count how many times each value shows up. The one with the highest count is the mode. If two values tie, you have two modes. If no value repeats, there is no mode.

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SHSAT Tip

When a question says "on average," it almost always means the mean. When it says "typical" or "middle," think median. Read carefully!

SECTION 5

When to Use Each Measure

Not every measure of center works equally well in every situation. Sometimes the mean gives a misleading picture because of an outlier (a number that is much larger or smaller than the rest). The chart below helps you pick the right tool.

The left box shows the original data set without an outlier, where the mean and median are equal at 10. The right box adds the outlier value of 50, which pulls the mean up to about 16.7 while the median barely changes to 11. The conclusion box explains that the median is the better choice when outliers are present.

Here's a quick rule of thumb: if the data has no outliers and is fairly symmetric, the mean is a great summary. If there are outliers or the data is skewed (lopsided), the median is more reliable. The mode is most useful when you care about the most popular or frequent value, like the most common shoe size sold at a store.

SECTION 6

Worked Example

Let's walk through a full SHSAT-style problem step by step.

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Problem

Class A quiz scores: 72, 85, 88, 90, 95. Class B quiz scores: 68, 78, 82, 92, 100. Which class performed better on average, and which class has the higher median?

Step-by-Step Solution

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Step 1 — Find the mean of Class A

Add all scores: 72 + 85 + 88 + 90 + 95 = 430. Divide by the number of scores: 430 ÷ 5 = 86.

Mean of Class A = 86
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Step 2 — Find the mean of Class B

Add all scores: 68 + 78 + 82 + 92 + 100 = 420. Divide by the number of scores: 420 ÷ 5 = 84.

Mean of Class B = 84
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Step 3 — Compare the means

Class A's mean (86) is greater than Class B's mean (84). So on average, Class A scored higher.

Class A has the higher mean.
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Step 4 — Find the median of each class

Both data sets have 5 values (odd count), so the median is the 3rd value when sorted. Class A is already sorted: the 3rd value is 88. Class B is already sorted: the 3rd value is 82.

Median of Class A = 88 | Median of Class B = 82
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Step 5 — Compare the medians

Class A's median (88) is also greater than Class B's median (82). Both the mean and median agree: Class A performed better.

Class A has the higher median (88 > 82).

SECTION 7

Strengths & Limitations of Each Measure

Each measure of center has strengths and weaknesses. Knowing these will help you pick the right one on test day — and explain your reasoning.

Measure	Strengths	Limitations
Mean	Uses every data value; great for symmetric data without outliers; most commonly used average.	Easily pulled toward outliers (extreme highs or lows); can be misleading for skewed data.
Median	Not affected by outliers; represents the true middle of the data; works well with skewed sets.	Ignores actual values above and below the middle; less useful for further calculations.
Mode	Easy to find; works for non-numerical data (like favorite color); shows the most popular value.	A data set may have no mode or many modes; doesn't use all values; not always near the center.

Comparing the three measures of center

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KEY TAKEAWAY

Choosing a measure of center is like choosing the right tool in a toolbox. A hammer is great for nails, but terrible for screws. The mean is your go-to for "normal" data, but switch to the median when outliers show up. The SHSAT often tests whether you know which tool to reach for.

SECTION 8

Connection to More Advanced Ideas

Once you're comfortable comparing data sets with measures of center, you're ready to start thinking about measures of spread (also called variability). Spread tells you how clustered or stretched out the data is. In high school, you'll learn about the range, interquartile range (IQR), and standard deviation.

What You Know Now	What Comes Next
Mean — the average of a data set	Weighted mean — when some values count more than others (like test grades worth different points)
Median — the middle value	Box-and-whisker plots — a picture that shows the median, quartiles, and outliers all at once
Comparing centers of two data sets	Comparing centers AND spreads together to make stronger conclusions about which data set is "better"
Spotting outliers by eye	Using formulas (like 1.5 × IQR) to officially identify outliers

Current knowledge vs. advanced topics

For now, focus on mastering the mean, median, and mode. These three measures form the foundation for everything else in statistics. Every advanced technique builds on the same basic question: What single number best represents this data set?

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Data Set X has a mean of 50 and a median of 50. Data Set Y has a mean of 50 and a median of 45. What can you conclude about Data Set Y compared to Data Set X?

Show answer

Both data sets have the same mean (50), so on average they are equal. However, Data Set Y's median (45) is lower than its mean, which tells us that Data Set Y is probably skewed to the right — it has some high values pulling the mean up above the middle. Data Set X, with equal mean and median, is more symmetric. So even though the averages match, the data is shaped differently.

Question 2: BASIC CALCULATION

Group A scores: 60, 70, 75, 80, 90. Group B scores: 65, 72, 78, 84, 86. Find the mean of each group and state which group has the higher average.

Show answer

Group A mean: (60 + 70 + 75 + 80 + 90) ÷ 5 = 375 ÷ 5 = 75. Group B mean: (65 + 72 + 78 + 84 + 86) ÷ 5 = 385 ÷ 5 = 77. Group B has the higher mean (77 > 75). Even though Group A has the single highest score (90), Group B's scores are more consistently high, which gives it a better average overall.

Question 3: INTERMEDIATE

Store A daily sales (in dollars): 120, 150, 155, 160, 200. Store B daily sales: 100, 140, 155, 170, 350. Find the mean and median for each store. Which measure better represents Store B's typical daily sales? Explain.

Show answer

Store A: Mean = (120 + 150 + 155 + 160 + 200) ÷ 5 = 785 ÷ 5 = 157. Median = 155 (the 3rd value). Store B: Mean = (100 + 140 + 155 + 170 + 350) ÷ 5 = 915 ÷ 5 = 183. Median = 155 (the 3rd value). Store B's mean (183) is much higher than its median (155) because of the outlier of 350. The median (155) better represents Store B's typical day because four out of five days had sales at or below 170. The 350 is an unusual spike that inflates the mean.

Question 4: APPLIED

A teacher recorded quiz scores for two periods. Period 1: 82, 85, 85, 88, 90, 92. Period 2: 78, 80, 85, 85, 85, 95. The teacher says Period 1 did better. Use the mean, median, and mode to either support or challenge this claim.

Show answer

Period 1: Mean = (82 + 85 + 85 + 88 + 90 + 92) ÷ 6 = 522 ÷ 6 = 87. Median = (85 + 88) ÷ 2 = 86.5. Mode = 85. Period 2: Mean = (78 + 80 + 85 + 85 + 85 + 95) ÷ 6 = 508 ÷ 6 ≈ 84.67. Median = (85 + 85) ÷ 2 = 85. Mode = 85. The teacher's claim is supported by the mean (87 > 84.67) and the median (86.5 > 85). Both modes are equal at 85, so the mode alone doesn't help distinguish the periods. Overall, Period 1 did perform better by two of the three measures.

Question 5: CRITICAL THINKING

Two data sets each have 6 values. Data Set P has a mean of 40 and a median of 42. Data Set Q has a mean of 42 and a median of 40. A student says: "Data Set Q is definitely higher because its mean is bigger." Is the student's reasoning complete? Explain what additional information you would want before deciding which data set is truly higher.

Show answer

The student's reasoning is not complete. The mean and median disagree about which data set is higher. Data Set Q has the higher mean (42 vs. 40), but Data Set P has the higher median (42 vs. 40). This crossover pattern suggests that Data Set Q might have an outlier pulling its mean up, while most of its values are actually lower. To make a better decision, you would want to see the actual data values, check for outliers, and look at the spread (like the range or IQR). You might also consider which measure is more appropriate given the context of the data. Simply picking the higher mean is not always the best approach.

SUMMARY

Lesson Summary

Summary

To compare data sets using measures of center, calculate the mean (add all values, divide by the count), the median (the middle value when data is sorted), and the mode (the most frequent value) for each data set. Then compare the same measure across data sets. The data set with the higher mean has a higher average, and the data set with the higher median has a higher typical middle value.

Remember that outliers can pull the mean away from the center, so the median is more reliable when extreme values are present. On the SHSAT, always check whether the question asks for the mean, median, or mode — and watch out for tricky data sets where these measures disagree. When in doubt, calculate more than one measure and compare them to get the full picture.