Grade: Grade 9 Subject: Mathematics Unit: Statistics SAT: ProblemSolving+DataAnalysis ACT: Math

Measures of Spread

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Measures of Spread (Variability)

Measures of spread describe how dispersed or spread out data values are from the center. While measures of center (mean, median, mode) tell us about typical values, measures of spread tell us about the consistency or variability in the data.

Understanding variability is essential for interpreting data. Two data sets can have the same mean but very different spreads, leading to very different conclusions.

Key Measures of Spread

Measure Definition When to Use
Range Maximum - Minimum Quick overview; sensitive to outliers
Interquartile Range (IQR) Q3 - Q1 (middle 50% of data) Resistant to outliers; use with median
Standard Deviation Average distance from the mean Most common; use with mean
Variance Standard deviation squared Used in advanced statistics

Range

Range = Maximum value - Minimum value

The simplest measure of spread, but easily affected by outliers.

Interquartile Range (IQR)

IQR = Q3 - Q1

Where Q1 is the 25th percentile (lower quartile) and Q3 is the 75th percentile (upper quartile).

The IQR represents the spread of the middle 50% of the data.

Standard Deviation

Understanding Standard Deviation

Standard deviation measures the typical distance of data points from the mean. A small standard deviation means data points cluster close to the mean; a large standard deviation means they are more spread out.

Calculating Standard Deviation (for a sample):

  1. Find the mean of the data
  2. Subtract the mean from each data point (find deviations)
  3. Square each deviation
  4. Find the average of the squared deviations (variance)
  5. Take the square root of the variance

SAT/ACT Connection

The SAT and ACT frequently test conceptual understanding of standard deviation rather than calculation. You should know: data with larger spread has larger standard deviation; adding a constant to all data doesn't change standard deviation; multiplying all data by a constant multiplies the standard deviation by that constant.

Examples

Work through these examples to understand measures of spread.

Example 1: Finding Range

Problem: Find the range of: 12, 18, 15, 22, 8, 19, 14

Step 1: Find the maximum value

Maximum = 22

Step 2: Find the minimum value

Minimum = 8

Step 3: Calculate range

Range = 22 - 8 = 14

Answer: The range is 14.

Example 2: Finding IQR

Problem: Find the IQR of: 3, 5, 7, 8, 12, 14, 16, 18, 20

Step 1: The data is already ordered. Find the median (Q2)

Median = 12 (middle value of 9 numbers)

Step 2: Find Q1 (median of lower half: 3, 5, 7, 8)

Q1 = (5 + 7)/2 = 6

Step 3: Find Q3 (median of upper half: 14, 16, 18, 20)

Q3 = (16 + 18)/2 = 17

Step 4: Calculate IQR

IQR = Q3 - Q1 = 17 - 6 = 11

Answer: The IQR is 11.

Example 3: Calculating Standard Deviation

Problem: Find the standard deviation of: 4, 6, 8, 10, 12

Step 1: Find the mean

Mean = (4 + 6 + 8 + 10 + 12)/5 = 40/5 = 8

Step 2: Find deviations from mean

4-8 = -4, 6-8 = -2, 8-8 = 0, 10-8 = 2, 12-8 = 4

Step 3: Square the deviations

16, 4, 0, 4, 16

Step 4: Find variance (average of squared deviations)

Variance = (16 + 4 + 0 + 4 + 16)/5 = 40/5 = 8

Step 5: Take square root for standard deviation

Standard deviation = sqrt(8) = 2.83

Answer: The standard deviation is approximately 2.83.

Example 4: Comparing Spreads

Problem: Set A: 48, 49, 50, 51, 52. Set B: 30, 40, 50, 60, 70. Both have mean 50. Which has greater spread?

By Range:

Set A: 52 - 48 = 4

Set B: 70 - 30 = 40

By Standard Deviation (conceptual):

Set A: values are very close to 50

Set B: values are far from 50

Answer: Set B has much greater spread. The range is 10 times larger, and the standard deviation would also be much larger.

Example 5: Effect of Outliers

Problem: Data: 10, 11, 12, 13, 14, 50. How do outliers affect range vs. IQR?

Range: 50 - 10 = 40 (heavily affected by outlier 50)

IQR:

Q1 = (11 + 12)/2 = 11.5

Q3 = (13 + 14)/2 = 13.5

IQR = 13.5 - 11.5 = 2

Answer: The range (40) is greatly inflated by the outlier, but the IQR (2) correctly shows that the middle 50% of data only spreads across 2 units. IQR is resistant to outliers.

Practice

Calculate or interpret measures of spread for these problems.

1. Find the range: 25, 32, 18, 45, 29, 33, 21

A) 18   B) 27   C) 45   D) 14

2. Find Q1 for: 2, 4, 5, 7, 8, 10, 12, 15

A) 4   B) 4.5   C) 5   D) 6

3. Find Q3 for: 2, 4, 5, 7, 8, 10, 12, 15

A) 10   B) 11   C) 12   D) 8

4. Using Q1 = 4.5 and Q3 = 11, find the IQR.

A) 6.5   B) 7.5   C) 15.5   D) 4.5

5. Which measure of spread is most affected by outliers?

A) IQR   B) Range   C) Median   D) Mode

6. Data set A has standard deviation 3. Data set B has standard deviation 8. Which statement is true?

A) Set A has more variability   B) Set B has more variability   C) Both have equal variability   D) Cannot determine

7. If 5 is added to every value in a data set, the standard deviation:

A) Increases by 5   B) Decreases by 5   C) Stays the same   D) Doubles

8. If every value in a data set is multiplied by 3, the standard deviation:

A) Stays the same   B) Is multiplied by 3   C) Is multiplied by 9   D) Is divided by 3

9. The five-number summary is: Min=5, Q1=12, Median=18, Q3=24, Max=35. What is the IQR?

A) 30   B) 12   C) 6   D) 18

10. Which data set has the largest standard deviation?

A) 50, 50, 50, 50   B) 49, 50, 50, 51   C) 40, 45, 55, 60   D) 48, 49, 51, 52

Click to reveal answers
  1. B) 27 - Range = 45 - 18 = 27
  2. B) 4.5 - Q1 is median of lower half (2,4,5,7): (4+5)/2 = 4.5
  3. B) 11 - Q3 is median of upper half (8,10,12,15): (10+12)/2 = 11
  4. A) 6.5 - IQR = Q3 - Q1 = 11 - 4.5 = 6.5
  5. B) Range - Range uses max and min, which are most affected by outliers
  6. B) Set B has more variability - Larger standard deviation = more spread
  7. C) Stays the same - Adding a constant shifts all values but doesn't change spread
  8. B) Is multiplied by 3 - Multiplying by k multiplies standard deviation by k
  9. B) 12 - IQR = Q3 - Q1 = 24 - 12 = 12
  10. C) 40, 45, 55, 60 - Values are most spread from their mean; other sets cluster more tightly

Check Your Understanding

Answer these reflection questions to deepen your understanding.

1. Why is IQR often preferred over range when describing spread?

Reveal Answer

IQR is preferred because it is resistant to outliers. The range only uses the two extreme values, so a single outlier can dramatically inflate the range and give a misleading picture of the data's spread. IQR focuses on the middle 50% of data, ignoring extreme values. This makes it more reliable for skewed distributions or data with outliers.

2. Two classes have the same mean test score of 75. Class A has a standard deviation of 5, and Class B has a standard deviation of 15. What does this tell you about the two classes?

Reveal Answer

Class A's scores are much more consistent - most students scored close to 75. Class B's scores are much more spread out - some students scored very high and some very low, even though the average is the same. In Class A, you'd expect most scores between about 70-80. In Class B, scores might range from 60 to 90 or even wider. The higher standard deviation in Class B indicates greater variability in student performance.

3. Explain why adding a constant to every data value doesn't change the standard deviation.

Reveal Answer

Standard deviation measures how far values are from the mean. When you add a constant to every value, both the individual values AND the mean shift by the same amount. The distances between values and the mean remain exactly the same. For example, if the data is 2, 4, 6 with mean 4, and we add 10, we get 12, 14, 16 with mean 14. The distances from the mean (2, 0, 2) are unchanged, so standard deviation is unchanged.

4. When would you choose to report the IQR and median rather than standard deviation and mean?

Reveal Answer

Use IQR and median when the data is skewed or contains outliers. The mean and standard deviation are both heavily influenced by extreme values, so they can be misleading for non-symmetric distributions. For example, reporting mean income might be skewed by a few millionaires; median income better represents the "typical" person. Similarly, IQR gives a more accurate picture of spread when outliers are present.

🚀 Next Steps

  • Review any concepts that felt challenging
  • Move on to the next lesson when ready
  • Return to practice problems periodically for review