Describe the merits and limitations of standard deviation. Compute standard deviation for the following data : 2, 12, 14, 17, 10, 9, 8, 4, 19, 4.

1. Merits of Standard Deviation

Standard deviation is a widely used measure of variability or dispersion in a data set. It offers several advantages:

a. Sensitivity to Variability: Standard deviation takes into account the differences between individual data points and the mean, providing a more accurate representation of the variability within the data set compared to simpler measures such as range or mean absolute deviation.

b. Reflects Spread of Data: Standard deviation provides information about the spread or distribution of data points around the mean. A higher standard deviation indicates greater variability, while a lower standard deviation suggests that data points are closer to the mean.

c. Useful in Comparing Samples: Standard deviation enables researchers to compare the variability of different data sets or samples. By calculating the standard deviation for each group, researchers can determine whether one group exhibits greater variability or dispersion compared to another.

d. Basis for Inferential Statistics: Standard deviation serves as a key component in many inferential statistical techniques, including hypothesis testing, confidence intervals, and analysis of variance (ANOVA). It helps assess the significance of differences between groups or the precision of estimates based on sample data.

2. Limitations of Standard Deviation

Despite its usefulness, standard deviation has certain limitations:

a. Affected by Outliers: Standard deviation is sensitive to extreme values or outliers in the data set. Outliers can disproportionately influence the calculation of standard deviation, leading to overestimation or underestimation of variability, particularly in small sample sizes.

b. Not Robust to Skewness: Standard deviation assumes that the distribution of data is symmetric and bell-shaped (i.e., normal distribution). In cases where the distribution is skewed or non-normal, standard deviation may not accurately reflect the spread of data or provide meaningful insights into variability.

c. Requires Numerical Data: Standard deviation can only be calculated for numerical data. It cannot be computed for categorical or ordinal data, limiting its applicability in certain contexts.

d. Dependent on Scale: Standard deviation is influenced by the scale or units of measurement used in the data set. Changes in the scale (e.g., converting from inches to centimeters) can alter the magnitude of the standard deviation without changing the underlying variability of the data.

3. Calculation of Standard Deviation

To compute the standard deviation for the given data set: 2, 12, 14, 17, 10, 9, 8, 4, 19, 4, we follow these steps:

a. Calculate the Mean:
Mean = (2 + 12 + 14 + 17 + 10 + 9 + 8 + 4 + 19 + 4) / 10
= 99 / 10
= 9.9

b. Calculate the Deviations from the Mean:
Deviation from mean for each data point: (-7.9, 2.1, 4.1, 7.1, 0.1, -0.9, -1.9, -5.9, 9.1, -5.9)

c. Square the Deviations:
Squared deviations: (62.41, 4.41, 16.81, 50.41, 0.01, 0.81, 3.61, 34.81, 82.81, 34.81)

d. Calculate the Variance:
Variance = (62.41 + 4.41 + 16.81 + 50.41 + 0.01 + 0.81 + 3.61 + 34.81 + 82.81 + 34.81) / 10
= 290.1 / 10
= 29.01

e. Calculate the Standard Deviation:
Standard deviation = √29.01
≈ 5.39

So, the standard deviation for the given data set is approximately 5.39.

Conclusion

Standard deviation is a valuable measure of variability that provides insights into the spread or dispersion of data points around the mean. While it offers several advantages, such as sensitivity to variability and usefulness in inferential statistics, standard deviation also has limitations, including sensitivity to outliers, dependence on data scale, and assumption of normal distribution. Understanding these merits and limitations is essential for accurate interpretation and meaningful application of standard deviation in data analysis and research.