Standadized normal distribution of statistics.pptx

standerdized normal distribution

1. Statistics

2. Variance Variance is a measurement of the spread between numbers in a data set.

3. Standard Deviation In statistics, the standard deviation is a measure of the amount of variation of a random variable expected about its mean . A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range

4. Raw Scores The definition of a raw score in statistics is an unaltered measurement. Raw scores have not been weighted, manipulated, calculated, transformed, or converted. An entire data set that has been unaltered is a raw data set.

5. Z - Score Z-score is a statistical measure that quantifies the distance between a data point and the mean of a dataset . It's expressed in terms of standard deviations. It indicates how many standard deviations a data point is from the mean of the distribution.

6. Z - Score For a recent final exam in STAT 500, the mean was 68.55 with a standard deviation of 15.45. If you scored an 80%: 𝑍=(80−68.55 )/15.45=0.74 , which means your score of 80 was 0.74 SD above the mean. If you scored a 60%: 𝑍=(60−68.55 )/15.45 =−0.55, which means your score of 60 was 0.55 SD below the mean.

7. Z - Score The scores can be positive or negative. For data that is symmetric (i.e. bell-shaped) or nearly symmetric, a common application of Z-scores for identifying potential outliers is for any Z-scores that are beyond ± 3.

8. Using z-scores to standardise a distribution Every X value in a distribution can be transformed into a corresponding z-score Any normal distribution can be standardized by converting its values into z scores.  Z scores tell you how many standard deviations from the mean each value lies . Converting a normal distribution into a z-distribution allows you to calculate the probability of certain values occurring and to compare different data sets

9. Using z-scores to make comparison we can compare performance [values] in two different distributions, based on their z-scores . Lower z-score means closer to the meanwhile higher means more far away . Positive means to the right of the mean or greater while negative means lower or smaller than the mean

10. Using z-scores to make comparison Jared scored a 92 on a test with a mean of 88 and a standard deviation of 2.7. Jasper scored an 86 on a test with a mean of 82 and a standard deviation of 1.8. Find the Z-scores for Jared's and Jasper's test scores, and use them to determine who did better on their test relative to their class.

11. Using z-scores to make comparison Step 1 : Compute each test score's Z-score using the mean and standard deviation for that test. For Jared's test, the Z-score is: 𝑍=(𝑥−𝜇)/𝜎 = (92−88)/2.7=4/2.7 = 1.48 For Jasper's test, the Z-score is: 𝑍=(𝑥−𝜇)/𝜎 = (86−82)/1.8 = 4/1.8 = 2.22

12. Using z-scores to make comparison Step 2 : Use Z-scores to compare across data sets. Jared's Z-score of 1.48 says that his score of 92 was between 1 and 2 standard deviations above the mean. Jasper's Z-score of 2.22 says that his score of 86 was a bit more than 2 standard deviations above the mean. So, Jasper's score of 86 was relatively higher for his class than Jared's 92 was for his class.

13. Probability Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.

14. What are Equally Likely Events? When the events have the same theoretical probability of happening, then they are called equally likely events. The results of a sample space are called equally likely if all of them have the same probability of occurring. For example, if you throw a die, then the probability of getting 1 is 1/6. Similarly, the probability of getting all the numbers from 2,3,4,5 and 6, one at a time is 1/6. Hence, the following are some examples of equally likely events when throwing a die : Getting 3 and 5 on throwing a die Getting an even number and an odd number on a die Getting 1, 2 or 3 on rolling a die are equally likely events, since the probabilities of each event are equal