How do you find the area under the standard normal curve between two z-scores, z1 and z2?

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To find the area under the standard normal curve between two z-scores, z1 and z2, you can follow these steps:

  1. Understand the Standard Normal Distribution: The standard normal distribution is a bell-shaped curve that represents the distribution of a variable that has been standardized to a mean of 0 and a standard deviation of 1. The z-scores represent the number of standard deviations a data point is from the mean.
  2. Use the Z-Table or Standard Normal Distribution Calculator: To find the area between z1 and z2, you will need to consult a z-table or use an online standard normal distribution calculator. These resources provide the cumulative area (or probability) to the left of a given z-score.
  3. Calculate the Area:
    1. Find the cumulative area for z1: Look up z1 in the z-table or calculator. This value represents the area to the left of z1.
    2. Find the cumulative area for z2: Similarly, find the cumulative area for z2. This value represents the area to the left of z2.
    3. Subtract the two areas: The area between z1 and z2 is calculated by subtracting the cumulative area at z1 from the cumulative area at z2:
      • Area = P(Z < z2) – P(Z < z1)
  4. Example: Suppose z1 = -1 and z2 = 1.
    1. Find P(Z < 1): From the z-table, this is approximately 0.8413.
    2. Find P(Z < -1): From the z-table, this is approximately 0.1587.
    3. Calculate the area: Area = 0.8413 – 0.1587 = 0.6826.
    4. Interpretation: This means that approximately 68.26% of the data lies between z = -1 and z = 1.

In summary, by using a z-table or a calculator, you can easily find the area under the standard normal curve between any two z-scores, allowing you to understand the probabilities associated with different segments of the distribution.

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