Averages and Uncertainty¶
Level: Middle School (Ages 13-15)
Introduction¶
You've learned about estimation and combining information. Now let's add some math to make our estimates even better! Don't worry - we'll start simple.
What is an Average?¶
An average (also called the mean) is a way to find the "middle" of a group of numbers.
Formula¶
Example 1: Test Scores¶
Your test scores: 85, 90, 88, 92, 85
Average = (85 + 90 + 88 + 92 + 85) ÷ 5 = 440 ÷ 5 = 88
Your average score is 88.
Example 2: Measuring Height¶
You measure your height 5 times: - 65.2 inches - 65.5 inches - 65.1 inches - 65.4 inches - 65.3 inches
Average = (65.2 + 65.5 + 65.1 + 65.4 + 65.3) ÷ 5 = 327.5 ÷ 5 = 65.5 inches
The average is probably closer to your true height than any single measurement!
Why Averages Matter for Estimation¶
When you have noisy measurements, the average is usually a better estimate than any single measurement.
The Dart Board Analogy¶
Imagine throwing 10 darts at a target: - Some land left of center - Some land right of center - Some land high - Some land low
The average position of all darts is probably very close to where you were aiming!
What is Uncertainty?¶
Uncertainty is how much you're not sure about something. It's like saying "I think it's 50, but I could be wrong by about 5."
Example: Guessing Jellybeans¶
You guess there are 100 jellybeans in a jar.
Low uncertainty: "I'm pretty sure it's between 95 and 105" High uncertainty: "It could be anywhere from 50 to 150"
Measuring Uncertainty: Variance¶
Variance measures how spread out numbers are from the average.
Example: Two Students¶
Student A's test scores: 88, 89, 90, 89, 89 - Average: 89 - Scores are very close together (low variance) - Consistent performance!
Student B's test scores: 70, 95, 85, 100, 95 - Average: 89 (same average!) - Scores are spread out (high variance) - Inconsistent performance!
Calculating Variance¶
Step 1: Find the average Step 2: Find how far each number is from the average Step 3: Square those differences Step 4: Average the squared differences
Example: Student A¶
Scores: 88, 89, 90, 89, 89 Average: 89
Differences from average: - 88 - 89 = -1 - 89 - 89 = 0 - 90 - 89 = 1 - 89 - 89 = 0 - 89 - 89 = 0
Squared differences: - (-1)² = 1 - (0)² = 0 - (1)² = 1 - (0)² = 0 - (0)² = 0
Variance = (1 + 0 + 1 + 0 + 0) ÷ 5 = 0.4
Example: Student B¶
Scores: 70, 95, 85, 100, 95 Average: 89
Differences: -19, 6, -4, 11, 6 Squared: 361, 36, 16, 121, 36
Variance = (361 + 36 + 16 + 121 + 36) ÷ 5 = 114
Student B has much higher variance (114 vs 0.4)!
Standard Deviation¶
Standard deviation is the square root of variance. It's easier to understand because it's in the same units as your measurements.
Student A: √0.4 = 0.63 points Student B: √114 = 10.68 points
Student B's scores vary by about 11 points, while Student A's vary by less than 1 point!
Uncertainty in Measurements¶
Example: Temperature Sensor¶
You measure temperature 10 times: 72, 73, 71, 72, 74, 72, 71, 73, 72, 71
Average: 72.1°F Variance: 1.09 Standard Deviation: 1.04°F
What this means: The temperature is probably 72.1°F, give or take about 1°F.
We can write this as: 72.1 ± 1.0°F
Combining Averages¶
When you have multiple measurements, you can combine them!
Example: Two Thermometers¶
Thermometer A (less accurate): - Measurements: 70, 72, 71, 73, 70 - Average: 71.2°F - Standard deviation: 1.3°F
Thermometer B (more accurate): - Measurements: 72.0, 72.1, 71.9, 72.0, 72.1 - Standard deviation: 0.08°F - Average: 72.02°F
Which should you trust more? Thermometer B! It has much lower uncertainty.
The 68-95-99.7 Rule¶
For normally distributed data (bell curve): - 68% of measurements fall within 1 standard deviation - 95% fall within 2 standard deviations - 99.7% fall within 3 standard deviations
Example: Height Measurements¶
Your average height: 65.5 inches Standard deviation: 0.2 inches
- 68% chance your true height is between 65.3 and 65.7 inches
- 95% chance it's between 65.1 and 65.9 inches
- 99.7% chance it's between 64.9 and 66.1 inches
Uncertainty Decreases with More Measurements¶
The more measurements you take, the more certain you become!
Formula¶
Example: Measuring a Table¶
One measurement: 48 ± 0.5 inches (uncertainty = 0.5) Four measurements: 48 ± 0.25 inches (uncertainty = 0.5 ÷ √4 = 0.25) Nine measurements: 48 ± 0.17 inches (uncertainty = 0.5 ÷ √9 = 0.17)
More measurements = more certainty!
Kalman Filter Connection¶
The Kalman filter keeps track of: 1. The estimate (like an average) 2. The uncertainty (like variance)
It updates both as it gets new information!
Practice Problems¶
Problem 1: Calculate Average and Variance¶
You measure the length of a pencil 5 times (in cm): 19.2, 19.5, 19.3, 19.4, 19.1
a) What's the average length? b) What's the variance? c) What's the standard deviation? d) Write the length as: average ± standard deviation
Problem 2: Comparing Uncertainty¶
Sensor A: Average = 50, Standard Deviation = 5 Sensor B: Average = 50, Standard Deviation = 2
Which sensor is more reliable? Why?
Problem 3: More Measurements¶
You measure something once and get 100 ± 10. If you take 4 measurements and average them, what's the new uncertainty?
Problem 4: Temperature Readings¶
Morning temperatures for a week: 65, 67, 66, 68, 65, 66, 67
a) What's the average temperature? b) What's the standard deviation? c) What temperature range contains 95% of the measurements?
Real World Applications¶
GPS Accuracy¶
Your phone's GPS might say: - "You are at this location ± 5 meters"
The ±5 meters is the uncertainty! Sometimes it's ±3 meters (more certain), sometimes ±20 meters (less certain).
Scientific Measurements¶
Scientists always report measurements with uncertainty: - "The speed of light is 299,792,458 ± 1 m/s" - "The patient's temperature is 98.6 ± 0.2°F"
Weather Forecasts¶
"High of 75°F" really means "probably between 73°F and 77°F"
The forecast has uncertainty!
Try It Yourself!¶
Experiment 1: Reaction Time¶
- Use an online reaction time test
- Take the test 20 times
- Calculate the average
- Calculate the standard deviation
- What's your reaction time ± uncertainty?
Experiment 2: Measuring Objects¶
- Measure the length of your desk 10 times
- Calculate average and standard deviation
- Measure it 10 more times
- Calculate the new average and standard deviation
- Did the uncertainty decrease?
Experiment 3: Comparing Sensors¶
- Use two different rulers to measure the same object
- Take 5 measurements with each
- Calculate average and standard deviation for each
- Which ruler is more consistent (lower standard deviation)?
Key Concepts¶
- Average = Best single estimate from multiple measurements
- Variance = How spread out the measurements are
- Standard Deviation = Square root of variance (easier to interpret)
- Uncertainty = How much you might be wrong
- More measurements = Less uncertainty
What's Next?¶
Now that we understand averages and uncertainty, the next chapter will teach us about weighted averages - how to combine measurements when some are more reliable than others!
Key Vocabulary - Mean/Average: Sum of values divided by count - Variance: Average of squared differences from mean - Standard Deviation: Square root of variance - Uncertainty: How much error might be in your estimate - Normal Distribution: Bell curve shape of many natural measurements