The men’s super combined skiing is an event in the Winter Olympics. It consists of two races, a downhill and a slalom. The mean times for each of the races at the 2010 Winter Olympics in Vancouver are given in the table below, along with the standard deviations. Bode Miller of the USA won the gold medal. He skied the slalom in 50.8 seconds and the downhill in 114.0 seconds.

Race

Mean

(seconds)

Stand. Dev. (seconds)

Slalom

52.7

1.5

Downhill

116.3

1.9

On which race did he do better compared with the competition?

To determine in which race Bode Miller performed better relative to the competition, we can use the **z-score**. The z-score measures how many standard deviations a particular value is away from the mean. The formula for the z-score is:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \(x\) is Bode Miller's time in the race.
- \(\mu\) is the mean time of the race.
- \(\sigma\) is the standard deviation of the race times.

### For the **Slalom** race:
- Bode Miller's time: \(x = 50.8\) seconds
- Mean time: \(\mu = 52.7\) seconds
- Standard deviation: \(\sigma = 1.5\) seconds

\[
z_{\text{slalom}} = \frac{50.8 - 52.7}{1.5} = \frac{-1.9}{1.5} = -1.27
\]

### For the **Downhill** race:
- Bode Miller's time: \(x = 114.0\) seconds
- Mean time: \(\mu = 116.3\) seconds
- Standard deviation: \(\sigma = 1.9\) seconds

\[
z_{\text{downhill}} = \frac{114.0 - 116.3}{1.9} = \frac{-2.3}{1.9} = -1.21
\]

### Interpretation:
- **Slalom**: \(z = -1.27\)
- **Downhill**: \(z = -1.21\)

Both z-scores are negative, which means Bode Miller performed better than the average skier in both races. However, the z-score for the slalom race is lower (further from zero), meaning he performed better in the **slalom** race relative to the competition compared to the downhill race.

Would you like more details on z-scores or any other aspect? Here are five related questions to expand on the topic:

1. How is the z-score used in other areas of sports analysis?
2. What is the interpretation of a z-score in statistics in general?
3. Can you explain the relationship between standard deviation and performance consistency?
4. How would Bode Miller’s performance compare if the standard deviations were different?
5. How are z-scores related to the concept of normal distribution?

**Tip:** A z-score allows you to compare data points from different distributions by standardizing their values!

The men’s super combined skiing is an event in the Winter Olympics. It consists of two races, a downhill and a slalom. The mean times for each of the races at the 2010 Winter Olympics in Vancouver are given in the table below, along with the standard deviations. Bode Miller of the USA won the gold medal. He skied the slalom in 50.8 seconds and the downhill in 114.0 seconds.

Race

Mean (seconds)

Stand. Dev. (seconds)

Slalom: 52.7, 1.5

Downhill: 116.3, 1.9

On which race did he do better compared with the competition?

Analyze Bode Miller's performance in the slalom and downhill races at the 2010 Winter Olympics using z-scores. Learn how to calculate the z-score, compare his times with the competition, and interpret his performance in each race relative to the mean times and standard deviations. Suitable for high school students learning statistics.

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