KURT
Returns a measure of how peaked or flat a distribution is.
Syntax:
KURT(number1, number2, ... number30)
number1 to number30 are up to 30 numbers or ranges/arrays containing numbers.
KURT returns the kurtosis, a measure of how peaked or flat a distribution is, relative to a normal distribution. Positive values indicate a relatively peaked distribution, and negative a relatively flat distribution. KURT calculates:
for the n >= 4 numbers having a standard deviation s > 0.
Example:
KURT(A1:A30)
returns the kurtosis of the numbers in A1:A30.
KURT(1, 3, 4, 5, 7)
returns 0.2, indicating that this (too small to be useful) distribution is slightly peaked.
Application:
Analyzing Student Test Scores
The KURT function measures the kurtosis of a dataset. Kurtosis describes the shape of a distribution's tails in relation to its central peak.
- Positive Kurtosis (Leptokurtic): A distribution with a high, narrow peak and long, fat tails. This means a larger proportion of the data is concentrated around the mean, with a significant number of outliers.
- Negative Kurtosis (Platykurtic): A distribution with a low, broad peak and short, thin tails. This indicates that the data is spread out more evenly, with fewer extreme values.
Let's imagine we are a school administrator analyzing the results of a standardized math test across two different classes: Class A and Class B. We want to see how the scores are distributed in each class.
Here is the data for the test scores (out of 100):
Table: Math Test Scores
Student ID | Class | Score | ||
|---|---|---|---|---|
A | B | C | ||
1 | 101 | A | 75 | |
2 | 102 | A | 78 | |
3 | 103 | A | 80 | |
4 | 104 | A | 82 | |
5 | 105 | A | 85 | |
6 | 106 | B | 60 | |
7 | 107 | B | 70 | |
8 | 108 | B | 80 | |
9 | 109 | B | 90 | |
10 | 110 | B | 100 |
Analyzing the KURT Function for Each Class
- Class A: The scores are 75, 78, 80, 82, and 85. These scores are tightly clustered around the average.
- Calculation: When we calculate the KURT value for this dataset, we get a negative number. Let's assume the KURT function returns a value of approximately -0.378.
- Interpretation: This negative kurtosis (platykurtic distribution) tells us that the scores are more spread out with a relatively flat peak compared to a normal distribution. There are no extreme high or low scores, and the students performed fairly consistently, with scores spread across the middle range.
- Class B: The scores are 60, 70, 80, 90, and 100. There's an average score of 80, but there are also very low (60) and very high (100) scores.
- Calculation: When we calculate the KURT value for this dataset, we get a positive number. Let's assume the KURT function returns a value of approximately -1.2.
- Interpretation: This is an even more negative kurtosis than Class A, meaning it is flatter and has more uniform tails. This indicates the scores are much more spread out, and the data is very uniformly distributed across the range. The distribution is much flatter than a normal distribution. This might suggest a greater variability in the students' performance, with scores evenly distributed from the lowest to the highest.
Conclusion
Using the KURT function, the school administrator can make data-driven decisions:
- For Class A: The negative kurtosis shows a consistent performance with scores clustered around the mean. The teacher might focus on pushing the entire class to the next level of understanding.
- For Class B: The even more negative kurtosis highlights a greater spread in scores. The teacher might need to implement a more differentiated approach, providing extra support for the struggling students and more advanced challenges for the high-achievers.
This example illustrates how the KURT function can provide a concise numerical summary of a dataset's shape, which in turn can reveal underlying characteristics that are not immediately obvious from just looking at the average score.
Result of KURT for Class A:
Result of KURT for Class B:
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