Across the USA, results for these exams are normally distributed. What does that mean and why is this the case?
If you were to create a histogram of all GRE scores, what would you expect the histogram to look like? Would it be symmetrical? Would it be bell shaped? How many modes would it likely have? Would it be skewed?
Suppose that the mean GRE score for the USA is 500 and the standard deviation is 75. Use the Empirical Rule (also called the 68-95-99.7 Rule) to determine the percentage of students likely to get a score between 350 and 650? What percentage of students will get a score above 500? What percentage of students will get a score below 275? Is a score below 275 significantly different from the mean? Why or why not?
Choose any GRE score between 200 and 800. Be sure that you do not choose a score that a fellow student has already selected. Using your chosen score, how many standard deviations from the mean is your score? (This value is called the z-value). Using the table above (or the z table in Doc Sharing), what percentage of students will likely get a score below this value? What percentage of students are likely to get a score above this value?
Histogram would also be normally distributed. How many modes does a normal distribution have?
Do you know the 68-95-99.7 rule? Approximately 68% of scores in normal distribution are within one standard deviation (34% on each side of the mean), 95% within 2 SD, and 99.7% within 3 SD.
Z = (score-mean)/SD
1. When we say that the results for exams are normally distributed across the USA, it means that the scores of the exams tend to follow a bell-shaped curve when plotted on a histogram. This distribution occurs because many factors can influence exam scores, such as innate ability, preparation, and randomness. The combination of these factors often leads to a distribution that closely resembles a normal distribution.
2. If we were to create a histogram of all GRE scores, we would expect it to be symmetrical and bell-shaped. The majority of scores would cluster around the mean score, with fewer scores occurring further away from the mean. The histogram would typically have one mode, which represents the most frequently occurring score. However, in some cases, there can be multiple modes if there are distinct groups of students with different performance levels. A normal distribution is not skewed, so the histogram should not show any significant skewness.
3. The Empirical Rule, or the 68-95-99.7 Rule, states that in a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Using this rule, we can determine the percentage of students likely to get a score between 350 and 650:
- From the mean to one standard deviation below the mean (from 500 - 75 to 500), we have 34% of the scores.
- From one standard deviation below the mean to one standard deviation above the mean (from 500 to 500 + 75), we have 68% of the scores.
- Combining these two ranges, we get a total of 34% + 68% = 102% of the scores. However, we should keep in mind that the Empirical Rule provides approximations, so it's not unusual to get a slightly higher percentage due to rounding or approximation errors. In this case, we can assume that the percentage is around 100%.
To find the percentage of students likely to get a score above 500, we can use the same logic:
- From one standard deviation above the mean to three standard deviations above the mean (from 500 + 75 to 500 + 3*75), we have 15.87% + 2.28% + 0.13% = 18.28% of the scores.
To find the percentage of students likely to get a score below 275, we need to calculate the range between the mean and three standard deviations below the mean:
- From the mean to three standard deviations below the mean (from 500 - 3*75 to 500 - 75), we have 0.13% + 2.28% + 15.87% = 18.28% of the scores.
A score below 275 is significantly different from the mean because it falls outside of the range within three standard deviations of the mean (500 - 3*75 to 500 + 3*75). As per the Empirical Rule, we expect approximately 99.7% of the scores to fall within this range. Therefore, a score below 275 is considered significantly different because it lies in the extreme tail of the distribution.
4. Choosing any GRE score between 200 and 800, let's say we choose a score of 600. To determine the number of standard deviations our chosen score is from the mean, we can use the formula:
z = (x - mean) / standard deviation
where x is the chosen score, mean is the mean GRE score (500 in this case), and the standard deviation is 75.
Using the given formula, the calculation becomes:
z = (600 - 500) / 75
z = 100 / 75
z = 1.33
Therefore, our chosen score of 600 is 1.33 standard deviations above the mean.
To determine the percentage of students likely to get a score below this value, we can refer to the z-table or the standard normal distribution table. From the z-table, we find that a z-value of 1.33 corresponds to an area of 0.9080, which means that approximately 90.80% of students are likely to get a score below 600.
Similarly, the percentage of students likely to get a score above this value would be the complement of the percentage below, which is approximately 100% - 90.80% = 9.20%. Therefore, around 9.20% of students are likely to get a score above 600.
1. Normally distributed means that the data follows a specific mathematical pattern called a normal distribution or a Gaussian distribution. In this distribution, the data is symmetrically distributed around the mean, forming a bell-shaped curve. It is commonly found in many natural and social phenomena, including test scores, heights, and IQ scores.
The reason why test scores like the GRE often follow a normal distribution is due to several factors. One of the main reasons is that the exams are designed to measure a wide range of abilities, and the population taking the exam usually exhibits a range of skills and knowledge. This leads to a diverse distribution of scores, with some students performing well, some performing poorly, and most falling somewhere in the middle.
Additionally, normal distribution can be observed when multiple factors contribute to a particular outcome. In the case of the GRE, various factors such as studying, preparation, test anxiety, or inherent abilities can all influence test scores, leading to a spread of scores that cluster around the mean.
2. If all GRE scores were plotted on a histogram, it would likely be symmetrical and bell-shaped. The symmetrical nature arises due to the normal distribution, where the majority of scores would fall around the mean, tapering off towards the extremes. This creates a bell shape on the histogram.
Since the distribution is symmetric, there would typically be one mode, representing the peak of the distribution where the most scores are concentrated. In a properly designed exam, the mode should be very close to the mean score, indicating that the majority of students performed around the average level.
The histogram might not show any significant skewness unless there are external factors that systematically affect the scores of certain groups. In a representative sample of test-takers, the distribution would likely be symmetrical without any noticeable skew.
3. The Empirical Rule, also known as the 68-95-99.7 Rule, states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation from the mean.
- Approximately 95% falls within two standard deviations.
- Approximately 99.7% falls within three standard deviations.
Using this rule, we can determine the percentage of students likely to get a score between 350 and 650, which falls within two standard deviations from the mean.
Percentage of students likely to get a score between 350 and 650:
= 95% - 68%
= 27%
To determine the percentage of students who will get a score above 500 (the mean), we need to find the percentage of data that falls outside one standard deviation from the mean, which is 68%.
Percentage of students likely to get a score above 500:
= 100% - 68%
= 32%
To calculate the percentage of students likely to get a score below 275, we need to consider the third standard deviation. However, we need to check if the score is significantly different from the mean.
To determine if a score below 275 is significantly different from the mean, we can compare it to the range within three standard deviations. The mean (500) minus three standard deviations (3 x 75) gives us the lower limit of the range, which is 275. Since the score falls on this boundary, we can consider it to be within the range and not significantly different from the mean.
Therefore, the percentage of students likely to get a score below 275 is 0%.
4. To find the number of standard deviations (z-value) from the mean for a chosen GRE score, we can use the formula:
z = (x - mean) / standard deviation
Let's say the chosen score is 600.
z = (600 - 500) / 75
z = 1.33
Using a standard normal distribution table or a z-table, we can determine the corresponding percentage of students likely to get a score below this value and above this value.
From the z-table, a z-value of 1.33 corresponds to approximately 90.89% of students likely to get a score below this value.
To find the percentage of students likely to get a score above this value, we can subtract this percentage from 100%.
Percentage of students likely to get a score above 600:
= 100% - 90.89%
= 9.11%