Features of
Normal Distribution
Cody Derr American College of Education July 21, 2024
A normal distribution will graphically represente a peak in the middle with the mean representing the center data point. Decreases in data will make up the shoulders with the smallest amounts on the upper and lower tails as shown in figure 1 below.
Explanation
Figure 1
Normal distribution is a theoretical concept that allows reserchers to make estimations with the data to predict and analysis using the concepts of mean and standard deviation (Jana & Chakraborty, 2022). Data with a normal distribution will produce a "bell shaped" curve with higher frequencies for data points toward the middle and lower frequencies toward either end (Keller, 2016).
Note: This figure was obtained from Wanger & Gillespie (2019).
Illustration
Figure 2
- The center of the bell curve representes the mean( ) of the data set.
- 68% of all data points will fall within 1 standard deviation of the mean.
- 27% of data points will fall within 2 standard deviations of the mean
- 54% of data points will fall within 3 standard deviations of the mean.
- 1% of data points will fall outside of 3 standard deviations of the mean.
Note: This figure 2 was obtained from American College of Education (2024).
Note: This figure was obtained from Wanger & Gillespie (2019).
The Peak of a Bell Curve
The height of a bell curve represents three data points that are all the equal in a normal distribution.
- Mean - The average of all data points
- Median- The middle value of all data points
- Mode- The data value that is represented the most in the full data set
Upper and Lower Tails
The tails on a normal distribution graph represent the data points that are at the two opposite ends. The highest and lowest data points will be found at the tails as these are not represented with high frequency.
Outliers in the data can skew the graph to the left or right. When higher amounts of outliers are represented on one side, the curve is skewed in that direction.
Note. This figure was obtained from Wanger & Gillespie (2019).
Note. This figure was obtained from Wanger & Gillespie (2019).
Z Score
A z score is a value that describes how many standard deviations an individual data point is from the mean value (Wagner & Gillespie,2019).
How to find the z score
(Raw score - Mean)
Standard Deviation
z =
- Positive (+) z scores represent the raw data is greater than the mean.
- Negative (-) z score preresents the raw data is below the mean.
- A z score of zero means the data is equal to the mean.
Example
A raw score of 80, mean value of 75 with an SD of 10.
z= (80-75)/10 z= .5
Example from Wagner & Gillespie (2019)
Probability
Probably is directly related to the standard deviation of a normal distribution graph. The theoretical percentage of data in a normal distribution can be used to predict the probability of a new data point.
- 1 Standard Deviation
- 68% of data points will fall within 1 SD of the mean.
- There is a 68% probability that a new data point will within 1 SD.
- 2 Standard Deviations
- 27% of data points will fall within 2 SD of the mean.
- There is a 27% probability that a new data point will fall within 2 SD.
- 3 Standard Deviations
- 4% of data points will fall within 3 SD
- There is a 4% probability that a new data point will within 3 SD.
- 1% of data will fall outside of 3 SD
Note: This figure 2 was obtained from American College of Education (2024).
Summary
Assuming a normal distribution of data is useful for many statistical applications such as making inferences or prediction. The use of standard deviations and probability can be used to make decisions even when sample size may be low. Graphically representing this data provides a quick visual for highlighting the mean data point and the tails while displaying where the majority of data points fall. A basic understanding of the bell curve allows predictions to be made about the likelihood that a new data point will fall within a particular standard deviation. It is important to understand that statistical outliers may skew the graph and will need to be considered when making decisions.
References
American College of Education (2024). Res6003 Applied Statistics: Module 2 [Normal
distribution]. Canvas. https://ace.instructure.com/courses/2017347/modules/items/3955080
Jana, N., & Chakraborty, A. (2022). Estimation of order restricted standard deviations of
normal populations with a common mean. Statistics, 56(4), 867–890. https://doi.org/10.1080/02331888.2022.2079126
Keller, D. (2016). Bell-shaped—the normal curve. In The Tao of Statistics: A Path to
Understanding (With No Math) ( Second ed., pp. 36-39). SAGE Publications, Inc., https://doi.org/10.4135/9781483397429
Wagner, III, W., & Gillespie, B. (2019). data frequencies and distributions. In Using and
Interpreting Statistics in the Social, Behavioral, and Health Sciences (Vol. 0, pp.46-51 ). SAGE Publications, Inc., https://doi.org/10.4135/9781071814284
Features of Normal Distribution
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Transcript
Features of
Normal Distribution
Cody Derr American College of Education July 21, 2024
A normal distribution will graphically represente a peak in the middle with the mean representing the center data point. Decreases in data will make up the shoulders with the smallest amounts on the upper and lower tails as shown in figure 1 below.
Explanation
Figure 1
Normal distribution is a theoretical concept that allows reserchers to make estimations with the data to predict and analysis using the concepts of mean and standard deviation (Jana & Chakraborty, 2022). Data with a normal distribution will produce a "bell shaped" curve with higher frequencies for data points toward the middle and lower frequencies toward either end (Keller, 2016).
Note: This figure was obtained from Wanger & Gillespie (2019).
Illustration
Figure 2
Note: This figure 2 was obtained from American College of Education (2024).
Note: This figure was obtained from Wanger & Gillespie (2019).
The Peak of a Bell Curve
The height of a bell curve represents three data points that are all the equal in a normal distribution.
Upper and Lower Tails
The tails on a normal distribution graph represent the data points that are at the two opposite ends. The highest and lowest data points will be found at the tails as these are not represented with high frequency.
Outliers in the data can skew the graph to the left or right. When higher amounts of outliers are represented on one side, the curve is skewed in that direction.
Note. This figure was obtained from Wanger & Gillespie (2019).
Note. This figure was obtained from Wanger & Gillespie (2019).
Z Score
A z score is a value that describes how many standard deviations an individual data point is from the mean value (Wagner & Gillespie,2019).
How to find the z score
(Raw score - Mean)
Standard Deviation
z =
Example
A raw score of 80, mean value of 75 with an SD of 10.
z= (80-75)/10 z= .5
Example from Wagner & Gillespie (2019)
Probability
Probably is directly related to the standard deviation of a normal distribution graph. The theoretical percentage of data in a normal distribution can be used to predict the probability of a new data point.
Note: This figure 2 was obtained from American College of Education (2024).
Summary
Assuming a normal distribution of data is useful for many statistical applications such as making inferences or prediction. The use of standard deviations and probability can be used to make decisions even when sample size may be low. Graphically representing this data provides a quick visual for highlighting the mean data point and the tails while displaying where the majority of data points fall. A basic understanding of the bell curve allows predictions to be made about the likelihood that a new data point will fall within a particular standard deviation. It is important to understand that statistical outliers may skew the graph and will need to be considered when making decisions.
References
American College of Education (2024). Res6003 Applied Statistics: Module 2 [Normal
distribution]. Canvas. https://ace.instructure.com/courses/2017347/modules/items/3955080
Jana, N., & Chakraborty, A. (2022). Estimation of order restricted standard deviations of
normal populations with a common mean. Statistics, 56(4), 867–890. https://doi.org/10.1080/02331888.2022.2079126
Keller, D. (2016). Bell-shaped—the normal curve. In The Tao of Statistics: A Path to
Understanding (With No Math) ( Second ed., pp. 36-39). SAGE Publications, Inc., https://doi.org/10.4135/9781483397429
Wagner, III, W., & Gillespie, B. (2019). data frequencies and distributions. In Using and
Interpreting Statistics in the Social, Behavioral, and Health Sciences (Vol. 0, pp.46-51 ). SAGE Publications, Inc., https://doi.org/10.4135/9781071814284