Handout 1: Summary Statistics for Quantitative Variables

Unit 1, Topic 1.7: Summary Statistics for Quantitative Variables

Overview

This lesson focuses on using numbers to summarize quantitative data (like test scores or heights) with measures of center and spread. These summaries help us understand the data’s typical value and how much it varies. Context, like who the data is from, matters because it explains what the numbers mean. For example, an average of "5 hours" could be sleep for kids or work for adults without context.

Quantitative data uses numbers to measure amounts (e.g., time, weight). We’ll calculate the mean (average), median (middle), range (high to low), interquartile range (middle spread), and standard deviation (average spread from the mean). These help us describe the data and check claims.

Assignment:

Part 1: Guided Practice Activity

Work on your own. Use the data below from 15 students (from a class survey). Calculate summary statistics and interpret them.

Data:

  • Test Scores: 55, 60, 65, 68, 72, 75, 78, 80, 82, 85, 87, 90, 92, 95, 100

Tasks:

  1. Calculating Center:
    • Find the mean (add all scores and divide by 15).
    • Find the median (sort and pick the middle value).
    • Write 1-2 sentences comparing them (e.g., "The mean is lower than the median, showing a low outlier pulls it down.").
    • Extra Practice: Use your own data (e.g., "Daily Steps: 4000, 4500…" from 5 days). Calculate mean and median.
  2. Calculating Variability:
    • Find the range (highest minus lowest).
    • Find the interquartile range (see below) (IQR: Q3 - Q1, where Q1 is the median of the lower half, Q3 of the upper half).
    • Estimate standard deviation (average distance from the mean, roughly half the range for small sets).
    • Write 1-2 sentences about spread (e.g., "The range is wide, showing big differences in scores.").
    • Extra Practice: Calculate range and IQR for your "Daily Steps" data.
  3. Reflection:
    • Write 2-3 sentences interpreting the statistics in context and discussing outlier resistance (e.g., "The mean of 78 suggests good performance, but 55 lowers it; the median of 80 is better for skewed data. IQR resists outliers, focusing on the middle 50%.").

Part 2: Independent Practice

Look at this data from a survey of 12 students:

  • Heights (cm): 140, 145, 150, 152, 155, 158, 160, 165, 168, 170, 175, 185

Tasks:

  • Calculate the mean and median.
  • Calculate the range, IQR, and estimate standard deviation.
  • Write 2-3 sentences interpreting the statistics to assess a claim (e.g., "The average height claim of 160 cm fits, but 185 skews the mean upward."), using context.
  • Extra Activity: Invent a dataset for 10 people (e.g., "Screen Time (hours): 2, 3…"). Calculate mean, median, range, and IQR, then interpret a claim like "Teens average 5 hours" with context.

Homework Assignment

  • Collect data from 5 people on a quantitative variable (e.g., minutes spent reading: 10, 20…). Calculate the mean, median, range, and IQR, then interpret the results with context to share next class.

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The interquartile range (IQR) measures how spread out the middle 50% of a dataset is. It shows where most of the data lies and helps identify variability while ignoring extreme values.

How It’s Calculated

  1. Order all data from smallest to largest.

  2. Find Q1 (the first quartile) — the value at the 25th percentile (where 25% of data falls below it).

  3. Find Q3 (the third quartile) — the value at the 75th percentile (where 75% of data falls below it).

  4. Subtract Q1 from Q3:

    «math»«semantics»«mrow»«mtext»IQR«/mtext»«mo»=«/mo»«mi»Q«/mi»«mn»3«/mn»«mo»§#8722;«/mo»«mi»Q«/mi»«mn»1«/mn»«/mrow»«annotation encoding=¨application/x-tex¨»\text{IQR} = Q3 - Q1«/annotation»«/semantics»«/math»

Example

If your data is:
2, 4, 5, 7, 8, 10, 12, 14, 15

  • Q1 = 5 (25th percentile)

  • Q3 = 12 (75th percentile)

  • IQR = 12 − 5 = 7

Why It’s Useful

  • It shows the range of the middle 50% — where most values lie.

  • It helps spot outliers:
    Any value below

    «math»«semantics»«mrow»«mi»Q«/mi»«mn»1«/mn»«mo»§#8722;«/mo»«mn»1.5«/mn»«mo»§#215;«/mo»«mi»I«/mi»«mi»Q«/mi»«mi»R«/mi»«/mrow»«annotation encoding=¨application/x-tex¨»Q1 - 1.5 \times IQR«/annotation»«/semantics»«/math»

    or above

    «math»«semantics»«mrow»«mi»Q«/mi»«mn»3«/mn»«mo»+«/mo»«mn»1.5«/mn»«mo»§#215;«/mo»«mi»I«/mi»«mi»Q«/mi»«mi»R«/mi»«/mrow»«annotation encoding=¨application/x-tex¨»Q3 + 1.5 \times IQR«/annotation»«/semantics»«/math»

    is considered an outlier.

In short, the IQR tells you how tightly or loosely clustered the central data is, without being affected by extreme highs or lows.

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Standard deviation (SD) measures how spread out all the data points are from the mean (average). It shows how much the values vary or deviate from the average value.

How It’s Calculated

  1. Find the mean (average) of all data.

  2. Subtract the mean from each data point to find each deviation.

  3. Square each deviation (to remove negatives).

  4. Find the average of those squared deviations.

    • For a population, divide by n.

    • For a sample, divide by n – 1.

  5. Take the square root of that average.

Example

Data: 4, 7, 10

  • Mean = (4 + 7 + 10)/3 = 7

  • Deviations: −3, 0, +3

  • Squared deviations: 9, 0, 9

  • Average of squares = 9

  • SD = √9 = 3

What It Tells You

  • Small SD → data points are close to the mean (less variation).

  • Large SD → data points are spread out (more variation).

Example Interpretation

  • Test scores with SD = 2 → most students scored near the average.

  • Test scores with SD = 15 → students’ scores vary widely.

So, while the IQR looks at the middle 50% of data, the standard deviation looks at all values and how far they tend to stray from the average.

ONE MORE EXAMPLE OF SD

Standard Deviation for the Example Numbers

Using the 10 numbers I provided (42, 58, 63, 71, 77, 84, 89, 92, 96, 105), I'll calculate the standard deviation step by step. As before, there are two versions:

  • Population standard deviation (σ): For the full dataset (divide by n=10).
  • Sample standard deviation (s): For estimating from a sample (divide by n-1=9).

Step 1: Calculate the Mean (μ)

Sum = 42 + 58 + 63 + 71 + 77 + 84 + 89 + 92 + 96 + 105 = 777. μ = 777 / 10 = 77.7.

Step 2: Calculate Deviations and Squared Deviations

For each value, subtract the mean, square it, and sum them up.

Value Deviation (x - μ) Squared Deviation
42 -35.7 1,274.49
58 -19.7 388.09
63 -14.7 216.09
71 -6.7 44.89
77 -0.7 0.49
84 6.3 39.69
89 11.3 127.69
92 14.3 204.49
96 18.3 334.89
105 27.3 745.29
Sum 0 3,376.01

Step 3: Calculate Variance

  • Population variance (σ²) = Sum of squared deviations / 10 = 3,376.01 / 10 = 337.601.
  • Sample variance (s²) = Sum of squared deviations / 9 = 3,376.01 / 9 ≈ 375.112.

Step 4: Standard Deviation (Square Root of Variance)

  • Population σ = √337.601 ≈ 18.37.
  • Sample s = √375.112 ≈ 19.37.

These numbers show a wide spread, notice the low 42 and high 105 pull things apart!

Last modified: Sunday, 12 October 2025, 8:15 PM