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A z-score is a helpful tool for determining how far an individual data point is from the mean in terms of standard deviations. It's like a translator, turning raw scores into a standardized form so we can use the normal distribution table. The formula to calculate a z-score is \( z = \frac{X - \mu}{\sigma} \), where \(X\) is the data value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Imagine you want to know how well a selling price or distance is aligned with the average values. If the z-score is positive, the data point is above the mean; if negative, it's below. For example, if a house price in our data has a z-score of 1.25, it means the price is 1.25 standard deviations above the average price. Comparing z-scores across different datasets makes it easy to see how data varies from the expected norms.

Standard deviation is a key measure in statistics, providing insight into data spread or variability. It tells how much the individual data points deviate from the mean. A small standard deviation means data points are close to the mean, while a large one indicates more spread.

The formula is given by \( \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(X_i - \mu)^2} \), where \(N\) is the number of data points, \(X_i\) are the values, and \(\mu\) is the mean.
- **High variability**: Wider range of values, hence a larger standard deviation.- **Low variability**: Data points clustered around the mean implies a smaller standard deviation.

In the context of real estate, knowing this variability helps predict future pricing and identify unusual selling prices or property distances from the city center. It’s like the compass in understanding how much the properties differ from the expected average.

Probability estimation involves calculating the likelihood of an event occurring within a specific range of the normal distribution. Once you identify the z-score of a particular value, you can use the standard normal distribution table to estimate the probability of a data point falling above or below a certain threshold.

Here's how you might apply this in different scenarios:

• Estimate how many homes lie within a certain distance range from the city center.
• Predict the number of homes selling for over a set price point.

By looking at the area under the curve of a normal distribution graph, we can infer probabilities. In Part a of our exercise, finding homes selling for over $280,000 involves determining the area to the right of the corresponding z-score on the graph. Subtracting the area to the left (found through the z-score probability) from 1 gives the sought-after probability.

Therefore, probability estimation is about harnessing the power of the normal distribution to make educated guesses about data behavior and tendencies.