Stat

• The mean is the average value.
• The total area under the normal curve is equal to 1
• Choose the appropriate operation based on step 2 and 3


  1. Use the table of the area under the normal curve to find the corresponding area

  2. Shade the region of the curve according to the condition of z-value whether it is below, above, or between

  3. Draw/sketch a normal curve and locate the given z-value on the normal curve
• Properties of Normal Curve


  • Bell-shaped
  • Curve is symmetrical about its center
  • Mean, median, and mode coincide at the center
  • Width of the curve is determined by the standard deviation of the distribution
• Example 1


  • Find the area to the left of −1.69. The area to the left of z= -1.69 is 0.0455
• Example 3

  • The heights of teachers in Sta. Catalina National High School are normally distributed with a mean of 150 cm and standard deviation of 15 cm. The height of Sir Victor has a z-score of 3.25. The actual height of Sir Victor is 197.5 cm
• Example 3


  • Find the area to the right of −1.35. The area to the right of z= -1.35 is 0.9115
• Example 1


  • Suppose IQ scores are normally distributed with a mean of 100 and standard deviation of 10. If your IQ is 85, what is your z-score? Your z-score is -1.50
• Example 5


  • Suppose you have the population values 40 and 70 and that their corresponding z-scores are -2 and 1, respectively. The mean and standard deviation of the population are 60 and 10, respectively
• Activity

  1. Use Empirical rule to complete the following table

  2. Write on the respective column the range or interval of the scores based on the given parameters

  3. Illustrate the distribution through a diagram
• Example scenarios

  • John scored at the 90th percentile in a unit test in science. The corresponding z-score of the 90th percentile is approximately 1.28
  • A score in the 96th percentile is above the majority of the data
  • The upper 10% of the normal curve corresponds to the top decile of the distribution
  • To obtain an A in the test, a minimum grade of approximately 84.08 is needed
  • The percentile rank of a score of 84 in a distribution with a mean of 80 and standard deviation of 15 is approximately 84.13%
• Example scenarios

  • (a) P(Z < 1.32)
  • (b) P(Z < -1.05)
  • (c) P(-0.75 < Z < 1.56)
  • (d) P(Z > -0.88)
• Label the shaded region and draw a conclusion

  1. When the z-value is to the left or any related terms (e.g. below, less than) just write the value we obtained in step 3

  2. When the z-value is to the right or any related terms (e.g. above, greater than), subtract 1 by the obtained value in step 3

  3. When the shaded region is in between of the two z-value, subtract the biggest by the smallest value obtained in step 3
• Example 2

  • On a nationwide placement test that is normally distributed, the mean was 125 and standard deviation was 15. If you scored 149, what was your z-score? Your z-score is 1.60
• Identifying regions under the normal curve corresponding to different standard normal values

  1. Convert a normal random variable to a standard normal variable and vice versa

  2. Compute probabilities and percentiles using the standard normal table
• Example 4

  • Find the area between z = -1.30 and z = 2. The area between z= -1.30 and z = 2 is 0.8804
• Example 2


  • Find the area to the left of −1.35. The area to the left of z= -1.35 is 0.0885
• Table of Areas under the Normal Curve

  • Also known as z-Table
  • This table gives the area of any value of z from -3.99 to 3.99
  • The value from this table will describe the area of the specific region of the curve to the left of the given z-value
• Converting a normal random variable to a standard normal variable and vice-versa

  1. Standardizing or Standardization is the procedure of converting a random variable x to a standard normal variable or z-score (standardized value)

  2. A z-score is a measure of the number of standard deviations (σ) a particular data value is away from the mean (μ)
• Standard normal distribution

  Defined as a normally distributed random variable having a mean of zero (μ= 0) and standard deviation of one
• Computing probabilities and percentiles using the standard normal table

  1. Percentile is a measure of relative standing

  2. A probability value corresponds to an area under the normal curve

  3. Find the probabilities for each of the following scenarios
• Example scenarios

  • (a) P(X < 19)
  • (b) P(10 < X < 19)
• Example 4

  • The time it takes for a cell to divide is normally distributed with an average of 60 minutes and standard deviation of 5 minutes. If its “mitosis” has a z-score of -1.35, it will take approximately 52.25 minutes
• Choose the appropriate operation based on step 2 and 3

  1. Use the table of the area under the normal curve to find the corresponding area

  2. Shade the region of the curve according to the condition of z-value whether it is below, above, or between

  3. Draw/sketch a normal curve and locate the given z-value on the normal curve
• Properties of Normal Curve

  • Bell-shaped
  • Curve is symmetrical about its center
  • Mean, median, and mode coincide at the center
  • Width of the curve is determined by the standard deviation of the distribution
• Example 1

  • Find the area to the left of −1.69. The area to the left of z= -1.69 is 0.0455
• The total area under the normal curve is equal to 1
• Activity

  1. Use Empirical rule to complete the following table

  2. Write on the respective column the range or interval of the scores based on the given parameters

  3. Illustrate the distribution through a diagram
• Example 3

  • Find the area to the right of −1.35. The area to the right of z= -1.35 is 0.9115
• Example 3

  • The heights of teachers in Sta. Catalina National High School are normally distributed with a mean of 150 cm and standard deviation of 15 cm. The height of Sir Victor has a z-score of 3.25. The actual height of Sir Victor is 197.75 cm
• Example 1

  • Suppose IQ scores are normally distributed with a mean of 100 and standard deviation of 10. If your IQ is 85, what is your z-score? Your z-score is -1.50
• Find the probabilities for each of the following

  • (a) P(Z < 1.32)
  • (b) P(Z < −1.05)
  • (c) P(−0.75 < Z < 1.56)
  • (d) P(Z > −0.88)
• Example 5

  • Suppose you have the population values 40 and 70 and that their corresponding z-scores are -2 and 1, respectively. The mean and standard deviation of the population are 60 and 10, respectively
• Identifying regions under the normal curve corresponding to different standard normal values

  1. Convert a normal random variable to a standard normal variable and vice versa

  2. Compute probabilities and percentiles using the standard normal table
• Table of Areas under the Normal Curve

  • Also known as z-Table
  • This table gives the area of any value of z from -3.99 to 3.99
  • The value from this table will describe the area of the specific region of the curve to the left of the given z-value
• Standard normal distribution

  Defined as a normally distributed random variable having a mean of zero (μ= 0) and standard deviation of one
• Label the shaded region and draw a conclusion

  1. When the z-value is to the left or any related terms (e.g. below, less than) just write the value we obtained in step 3

  2. When the z-value is to the right or any related terms (e.g. above, greater than), subtract 1 by the obtained value in step 3

  3. When the shaded region is in between of the two z-value, subtract the biggest by the smallest value obtained in step 3
• Example 4

  • Find the area between z = -1.30 and z = 2. The area between z= -1.30 and z = 2 is 0.8804
• Example 2

  • On a nationwide placement test that is normally distributed, the mean was 125 and standard deviation was 15. If you scored 149, what was your z-score? Your z-score is 1.60
• Example 2


  • Find the area to the left of −1.35. The area to the left of z= -1.35 is 0.0885