Answer to Question #99704 in Statistics and Probability for Nishit Tuli

Question

For the data in the table below:
i. What do high kurtosis and skewness figures denote?
ii. Free Sulphur dioxide and total Sulphur dioxide have mean, median and modes that
differ substantially from each other. What can you infer from this?
iii. By looking at the descriptive stats of the variables, identify two that may be closest to
being normally distributed. Why?
Desc Stats fixed acidity volatile acidity citric acid residual sugar chlorides free sulfur dioxide total sulfur dioxide density pH sulphates alcohol
Mean 6.85 0.28 0.33 6.39 0.05 35.31 138.36 0.99 3.19 0.49 10.51
Med 6.80 0.26 0.32 5.20 0.04 34.00 134.00 0.99 3.18 0.47 10.40
Mod 6.80 0.28 0.30 1.20 0.04 29.00 111.00 0.99 3.14 0.50 9.40
SD 0.84 0.10 0.12 5.07 0.02 17.01 42.50 0.00 0.15 0.11 1.23
Kurt 2.17 5.09 6.17 3.47 37.56 11.47 0.57 9.79 0.53 1.59 -0.70
Skew 0.65 1.58 1.28 1.08 5.02 1.41 0.39 0.98 0.46 0.98 0.49
Range 10.40 1.02 1.66 65.20 0.34 287.00 431.00 0.05 1.10 0.86 6.20

Answer 1

i) If the data sets have high kurtosis, that means the data set tends to have heavy tails or outliers. Kurtosis enables us to have an idea about the 'flatness or peakedness' of the frequency curve. Skewness is the measure of asymmetry in the distribution of sample data. So higher skewness shows more asymmetry in the distribution.

ii) Substantial difference in mean, median and the modes denotes that the data distribution is not symmetrical. It can be seen that for both the free sulphur dioxide and total sulphur dioxide $mode<median<mean$ which denotes that the data are rightly skewed not symmetrical and so their distribution will have a long tail in the right side of the number line.

iii) A data distribution is said to be normally distributed or symmetrical if the mean, median and the mode of that data distribution is almost similar or equal to each other.

From the data given, we can see that the mean, median and the mode of chlorides are $0.05,0.04$ and $0.04$ respectively with the standard deviation of 0.02 which shows that the distribution is very close to normal distribution.

Similarly, we can see that the mean, median and the modes for the density data are $0.99,0.99$ and $0.99$ respectively with the standard deviation of 0.00, so from here also, we can conclude that the distribution of density data will also follows normal distribution.