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What score indicators can be seen in the post-examination class report academic overview... expand

Zeng Liuchen

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Here, I will introduce to you the following six commonly used test data indicators.

1. Average score

In the same exam, the sum of the test scores is divided by the number of scores. The average score is the balanced score that balances all score deviations on both sides.

Average score = sum of all scores of candidates/number of candidates

Note features: Average score is very sensitive to the existence of very large or very small data in the examination data. The existence of extremely large or extremely small data will significantly change the average score of the test data.

2. Median

Arrange a set of test data from small to large, and the number in the middle is the median. (If there is an even number of test data, the average of the two middle values ??is usually taken as the median)

In the test data, if there are extreme data, the median can be used to understand the overall situation of the class. (Like the example above)

We can also understand the overall distribution tendency of the data from the deviation of the median from the mean.

For example: Class A took a certain mid-term exam, the class average score was 85 points, and the median score was 96 points. There are no extreme data. Please describe the overall situation of this class exam?

The median score of Class A in this test is 96, which is much higher than the class average score of 85. The overall class score is obviously negatively skewed, indicating that there are many people with low scores in this test.

The average score and median of the above two indicators are usually used to describe the central tendency of the exam. The mean is easily affected by a few extreme values; the median is the bisector of the frequency of the sample data, and it will not be affected by a few extreme values.

However, even if the two sets of test data have the same central tendency and the same average score, the changes in the scores may be very different. In order to describe the differences between data, statistics uses dispersion indicators to describe the degree of dispersion of data.

3. Standard deviation

Standard deviation is used to describe the distance between all observed values ??and the mean. The greater the change in a set of data, the greater its standard deviation.

Standard deviation formula

It can be seen from the standard deviation formula that the standard deviation reflects the sum of the distances between the observed values ??and the mean. When the data contains extreme values, the standard deviation is easily affected by it. .

Can the standard deviation of two sets of data be used for comparison? Generally speaking, two sets of data with the same mean can compare their standard deviations. The smaller the standard deviation, the smaller the dispersion of the set of data, that is, the smaller the fluctuation.

4. Interquartile range

When standard deviation is used to describe the dispersion of data, if it is affected by extreme values, the standard deviation will deviate. How to avoid it being affected by extreme values? In statistics, we can use the interquartile range to describe the dispersion of a data set.

The interquartile range (IQR) is a measure based on percentiles. Arrange a set of data in ascending order from small to large and divide it into three quartiles. The median is recorded as the second quartile Q2. The median of the data below the median is recorded as the first quartile Q1. The median of the above part of the data is recorded as the third quartile and recorded as Q3.

Figure: Interquartile range under normal distribution

The interquartile range IQR=Q3-Q1, the interquartile range gives the distance between 50% of the observed values, four The larger the quantile range is, the more scattered the observations are.

Example: In a large-scale mathematics test, the full score is 150 points, the average score is 107 points, the second quartile median is 109 points, the first quartile Q1: 94 points, the third quartile Q3: 120 points, describe the performance of Class A.

From the data, the average score of this class is 107 points, the class average score is slightly lower than the median, and there is no obvious abnormal data in the class. In this exam, 50% of the students scored between 94 and 120 points, with an interquartile range of 26 points.

The average and median describe the central tendency of the test data, and the standard deviation and interquartile range describe the degree of dispersion of the test data. However, even these four indicators cannot fully describe the test data. distributed. Here we introduce to you two commonly used indicators to describe data distribution, skewness and kurtosis.

5. Skewness

The skewness in statistics describes the symmetry of the data distribution.

The skewness index is equal to 0, the data is normally distributed, and the mean and median are equal. The distribution of high-scoring and low-scoring students in the class is similar.

The skewness index is greater than 0, the data is positively skewed (right-skewed), and the mean score is greater than the median. If the average score is much greater than the median, it means there are more high-scoring students in the class.

The skewness index is less than 0, the data is negatively skewed (left skewed), and the mean score is smaller than the median. If the average score is much smaller than the median, it means there are more people in the class with low scores.

Generally, when the skewness index is between -0.5 and +0.5, we consider the data to be normally distributed.

6. Kurtosis

Kurtosis describes how closely the data is distributed around the mean score.

Kurtosis equal to 3 is considered to be normal kurtosis. Generally speaking, normally distributed data is normal kurtosis, and the distribution of high-scoring and low-scoring people is equivalent.

When the kurtosis is less than 3, it is considered to be low kurtosis, with less data in the middle and high data at both ends. If the class scores show this peak, it means that the polarization of class scores is serious, and teachers should investigate the reasons themselves.

When the kurtosis is greater than 3, it is a peak-kurtosis state, and only a part of the data is distributed in a small number of positions, and the positions are random. If the class scores are in a peak-kurtosis state in the low-scoring position, it means that a small number of students in the class have scores in the low-scoring range. The teacher should investigate the reasons and understand the actual situation of these students and whether there are any problems with the teaching atmosphere.