Practice Noise in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Noise is random variation in data that is not explained by the underlying pattern or model — the unpredictable fluctuations around the true signal.

The static on a radio—it's there, but it's not the music you want to hear.

Showing a random 20 of 50 problems.

Example 1

medium
A poll of 50 people shows 52% support; a week later 48%. The true value is steady. Is the 4-point swing likely real change or noise?

Example 2

easy
Static on a radio that obscures the music is an analogy for what data concept?

Example 3

medium
A site averages 1000 visitors/day with day-to-day SD ≈50\approx 50. One day there are 10801080. Roughly how many standard deviations above the mean?

Example 4

easy
A thermometer in a stable environment reads: {20.1,19.9,20.2,20.0,19.8}\{20.1, 19.9, 20.2, 20.0, 19.8\}°C. The true temperature is 20°C. Calculate the noise (variability) and explain how taking more measurements would help.

Example 5

medium
Daily sales rise steadily from 100100 to 200200 over 100 days, with small day-to-day jitter. What is the signal and what is the noise?

Example 6

medium
A neighborhood records monthly burglaries: 4,6,3,5,7,5,4,64, 6, 3, 5, 7, 5, 4, 6. Is the spike of 7 likely signal or noise?

Example 7

medium
Stock prices show daily fluctuations. Stock A has daily changes: {+2%,−1%,+3%,−2%,+1%}\{+2\%, -1\%, +3\%, -2\%, +1\%\}. Stock B changes: {+0.1%,−0.1%,+0.1%,0%,+0.1%}\{+0.1\%, -0.1\%, +0.1\%, 0\%, +0.1\%\}. Identify which has more noise and what that means for investors.

Example 8

hard
In a medical trial, the treatment shows an improvement of 3 points on a pain scale, but each patient's response varies with SD = 8. With n=25n = 25 patients, calculate the signal-to-noise ratio (SNR = effect/SE) and determine if the signal can be detected.

Example 9

easy
Noise in data is best described as what?

Example 10

easy
To reduce noise in a length measurement, should you take more measurements or rely on a single reading?

Example 11

challenge
You fit polynomials of degree 1, 5, and 15 to 20 noisy data points generated by a roughly linear true relationship. Which model best generalizes to new data, and why?

Example 12

easy
Is all variation in data necessarily noise?

Example 13

medium
A dataset's variation comes partly from real group differences and partly from random measurement error. Which part is the noise?

Example 14

challenge
Averaging nn independent noisy measurements reduces the noise's spread by a factor of about n\sqrt{n}. By roughly what factor does the noise spread drop if you average 100 measurements instead of 1?

Example 15

medium
In a linear regression, what name is given to the difference between an observed yy and the predicted y^\hat y?

Example 16

medium
Monthly sales: 100,102,98,101,99100,102,98,101,99 around a flat average of 100100. Are the ups and downs better described as signal or noise?

Example 17

medium
Two thermometers in the same room read: A 20.0,20.0,20.0,20.020.0,20.0,20.0,20.0; B 19.5,20.4,20.1,19.919.5,20.4,20.1,19.9. Which is noisier?

Example 18

medium
A model perfectly predicts past training data but fails badly on new data. What likely went wrong?

Example 19

medium
Why does smoothing (e.g., a moving average) help reveal a trend in noisy data?

Example 20

easy
A stock rises three days in a row. A trader claims a guaranteed upward trend. Could this be noise?
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