Using a z-table to solve normal distribution problems

Percent to value steps - examples | value to percent steps - examples | abbreviated steps

We used a five step process in class to solve
normal distribution problems using a z-table.
Depending on the input given and the desired output, we worked the steps from 1 to 5 or 5 to 1.

In both cases, we started by drawing a sketch of a normal distribution with the mean and points of interest labeled.
This helped us get our bearings on the problem.  We returned to the sketch at the end of the problem to be sure to give an appropriate answer.

Here were our steps when given a percent and asked to find values:
  1. Use the problem to sketch a graph of the normal distribution involved
  2. Find the desired percent on the graph
  3. Find an approximation of this percent in the body of the z-table
  4. Find the corresponding z-score using the row and column titles on the z-table
  5. Find the corresponding deviation by multiplying the z score by the standard deviation
  6. Calculate the appropriate value(s) using the deviation and the mean (using addition or subtraction)
  7. Revisit the graph and the problem to apply and adjust your value(s) to answer the question
Here were our steps when given a value or values and asked to find a percent:
  1. Use the problem to sketch a graph of the normal distribution involved
  2. Calculate the appropriate deviation using the value(s) and the mean (using subtraction)
  3. Find the corresponding z-score by dividing the deviation by the standard deviation
  4. Find an approximation of this z-score using the row and column titles on the z-table
  5. Find the corresponding percent in the body of the z-table
  6. Label the desired percent on the graph
  7. Revisit the graph and the problem to apply and adjust your percent to answer the question
Here is an abbreviated version of the steps like we used in class:

Solve the following problems using a mean of 18 and a standard deviation of 5:

example 1 | example 2 | example 3

  1. Use the problem to sketch a graph of the normal distribution involved
  2. Find the desired percent on the graph
    1. Since we are looking for the middle 40%, and the z-table only shows information for the area between the mean and a point to the right of the mean, the desired percent on the graph is 20% (which represents the right half of the middle 40%)
  3. Find an approximation of this percent in the body of the z-table
    1. 20% can be written as .2000
    2. The closest value to .2000 in the body of the z-table is .1985
  4. Find the corresponding z-score using the row and column titles on the z-table
    1. The z-score that corresponds to .1985 is .52
  5. Find the corresponding deviation by multiplying the z score by the standard deviation
    1. .52*5 = 2.6 (5 is the standard deviation given for this problem)
  6. Calculate the appropriate value(s) using the deviation and the mean (using addition or subtraction)
    1. Since we want the scores that bound the middle 40% of the data, we want to add 2.6 to the mean to find the upper bound and subtract 2.6 from the mean to find the lower bound.
    2. 18 - 2.6 = 15.4 (18 is the mean given for this problem)
    3. 18 + 2.6 = 20.6 (18 is the mean given for this problem)
  7. Revisit the graph and the problem to apply and adjust your value(s) to answer the question
    1. The middle 40% of the data fall between 15.4 and 20.6
  1. Use the problem to sketch a graph of the normal distribution involved
  2. Find the desired percent on the graph
    1. Since we are looking for the third quartile, and the z-table only shows information for the area between the mean and a point to the right of the mean, the desired percent on the graph is 25% (which represents the area between the mean at 50% and the third quartile at 75%)
  3. Find an approximation of this percent in the body of the z-table
    1. 25% can be written as .2500
    2. The closest value to .2500 in the body of the z-table is .2486
  4. Find the corresponding z-score using the row and column titles on the z-table
    1. The z-score that corresponds to .2486 is .67
  5. Find the corresponding deviation by multiplying the z score by the standard deviation
    1. .67*5 =3.35 (5 is the standard deviation given for this problem)
  6. Calculate the appropriate value(s) using the deviation and the mean (using addition or subtraction)
    1. Since we want the third quartile, we want to add 3.35 to the mean.
    2. 18 + 3.35 = 21.35 (18 is the mean given for this problem)
  7. Revisit the graph and the problem to apply and adjust your value(s) to answer the question
    1. The third quartile is located at 21.35
  1. Use the problem to sketch a graph of the normal distribution involved
  2. Find the desired percent on the graph
    1. Since we are looking for the bottom 20%, and the z-table only shows information for the area between the mean and a point to the right of the mean, the desired percent on the graph is 30% (which represents the area between the mean at 50% and the upper 20%).  The distance between the mean and the upper 20% is the same as the distance between the mean and the lower 20%.
  3. Find an approximation of this percent in the body of the z-table
    1. 30% can be written as .3000
    2. The closest value to .3000 in the body of the z-table is .2995
  4. Find the corresponding z-score using the row and column titles on the z-table
    1. The z-score that corresponds to .2995 is .84
  5. Find the corresponding deviation by multiplying the z score by the standard deviation
    1. .84*5 =4.2 (5 is the standard deviation given for this problem)
  6. Calculate the appropriate value(s) using the deviation and the mean (using addition or subtraction)
    1. Since we want the lower 20%, we want to subtract 4.2 from the mean.
    2. 18 - 4.2 = 13.8 (18 is the mean given for this problem)
  7. Revisit the graph and the problem to apply and adjust your value(s) to answer the question
    1. 20% of the results fell below 13.8

Solve the following problems using a mean of 25 and a standard deviation of 6:

example 1 | example 2 | example 3 | example 4
        The second half of the problem involves the percentage between the mean (25) and the second value (40)
  1. Use the problem to sketch a graph of the normal distribution involved
  2. Calculate the appropriate deviation using the value(s) and the mean (using subtraction)
    1. 40 - 25 = 15 (25 is the mean given for this problem)
  3. Find the corresponding z-score by dividing the deviation by the standard deviation
    1. 15/6 = -2.5 (6 is the standard deviation given for this problem)
  4. Find an approximation of this z-score using the row and column titles on the z-table
    1. The z-score of 2.50 is precisely found on the table
  5. Find the corresponding percent in the body of the z-table
    1. A z-score of 2.50 corresponds to .4938 or 49.38%
  6. Label the desired percent on the graph
  7. Revisit the graph and the problem to apply and adjust your percent to answer the question

    Since the question asked for the area between 10 and 40, we add
    49.38% to 49.38% and get 98.76% for our final answer.
  1. Use the problem to sketch a graph of the normal distribution involved
  2. Calculate the appropriate deviation using the value(s) and the mean (using subtraction)
    1. 20 - 25 = -5 (25 is the mean given for this problem)
  3. Find the corresponding z-score by dividing the deviation by the standard deviation
    1. -5/6 = -.833... (6 is the standard deviation given for this problem)
  4. Find an approximation of this z-score using the row and column titles on the z-table
    1. The z-score of 0.83 is found on the table
  5. Find the corresponding percent in the body of the z-table
    1. A z-score of 0.83 corresponds to .2967 or 29.67%
  6. Label the desired percent on the graph
  7. Revisit the graph and the problem to apply and adjust your percent to answer the question

    Since the question asked for the area below 20, and the z-table gives the area between 20 and 25 (the mean), we subtract
    29.67% from 50% (the lower half) and get 20.33% for our final answer.
  1. Use the problem to sketch a graph of the normal distribution involved
  2. Calculate the appropriate deviation using the value(s) and the mean (using subtraction)
    1. 15 - 25 = -10 (25 is the mean given for this problem)
  3. Find the corresponding z-score by dividing the deviation by the standard deviation
    1. -10/6 = -1.666... (6 is the standard deviation given for this problem)
  4. Find an approximation of this z-score using the row and column titles on the z-table
    1. The z-score of 1.67 is found on the table
  5. Find the corresponding percent in the body of the z-table
    1. A z-score of 1.67 corresponds to .4525 or 45.25%
  6. Label the desired percent on the graph
  7. Revisit the graph and the problem to apply and adjust your percent to answer the question

    Since the question asked for the area above, and the z-table gives the area between 15 and 25 (the mean), we add
    45.25% to 50% (the upper half) and get 95.25% for our final answer.
  1. Use the problem to sketch a graph of the normal distribution involved
  2. Calculate the appropriate deviation using the value(s) and the mean (using subtraction)
    1. 22 - 25 = -3 (25 is the mean given for this problem)
  3. Find the corresponding z-score by dividing the deviation by the standard deviation
    1. -3/6 = -0.5 (6 is the standard deviation given for this problem)
  4. Find an approximation of this z-score using the row and column titles on the z-table
    1. The z-score of 0.50 is precisely found on the table
  5. Find the corresponding percent in the body of the z-table
    1. A z-score of 0.50 corresponds to .1915 or 19.15%
  6. Label the desired percent on the graph
        The second half of the problem involves the percentage between the mean (25) and the second value (35)
  1. Use the problem to sketch a graph of the normal distribution involved
  2. Calculate the appropriate deviation using the value(s) and the mean (using subtraction)
    1. 35 - 25 = 10 (25 is the mean given for this problem)
  3. Find the corresponding z-score by dividing the deviation by the standard deviation
    1. 10/6 = 1.666... (6 is the standard deviation given for this problem)
  4. Find an approximation of this z-score using the row and column titles on the z-table
    1. The z-score of 1.67 is found on the table
  5. Find the corresponding percent in the body of the z-table
    1. A z-score of 1.67 corresponds to .4525 or 45.25%
  6. Label the desired percent on the graph
  7. Revisit the graph and the problem to apply and adjust your percent to answer the question

    Since the question asked for the area between 22 and 35, we add
    19.15% to 45.25% and get 64.40% for our final answer.