Question

The performance of four students in annual report is given below: $$ \begin{array}{ccc} \hline \begin{array}{c} \text { Name of the } \\ \text { student } \end{array} & \begin{array}{c} \text { Mean score } \\ (\overline{\mathbf{x}}) \end{array} & \text { S.D. }(\sigma) \\ \hline \text { Dheeraja } & 75 & 11.25 \\ \text { Nishitha } & 65 & 5.98 \\ \text { Sindhuja } & 48 & 8.88 \\ \text { Akshitha } & 44 & 5.28 \\ \hline \end{array} $$ Who is less consistent than the others? (1) Dheeraja (2) Nishitha (3) Sindhuja (4) Akshitha

Short answer

Answer: Dheeraja

Step-by-step solution

Identify the data given

In this exercise, we are given the mean score (\(\overline{\mathbf{x}}\)) and standard deviation (\(\sigma\)) for each student. Dheeraja: \(\overline{\mathbf{x}} = 75\), \(\sigma = 11.25\) Nishitha: \(\overline{\mathbf{x}} = 65\), \(\sigma = 5.98\) Sindhuja: \(\overline{\mathbf{x}} = 48\), \(\sigma = 8.88\) Akshitha: \(\overline{\mathbf{x}} = 44\), \(\sigma = 5.28\)

Compare the standard deviations

Now let's compare the standard deviations to identify the least consistent student: Dheeraja: \(\sigma = 11.25\) Nishitha: \(\sigma = 5.98\) Sindhuja: \(\sigma = 8.88\) Akshitha: \(\sigma = 5.28\) From these values, we can see Dheeraja has the highest standard deviation (\(\sigma = 11.25\)).

Identify the least consistent student

Since Dheeraja has the highest standard deviation, she demonstrates the least consistency among the four students. Therefore, the answer is: (1) Dheeraja

Key concepts

Mean Score

The mean score is a fundamental concept in statistics. It represents the average score of a set of values which gives us a central value of those data points. In this context, the mean score of each student indicates their average performance in their assessments.
To find the mean, you sum all the scores and then divide by the total number of scores. It is represented by the symbol \( \overline{x} \).
Understanding the mean score helps us to quickly evaluate and compare the general level of performance between students.
This can be especially useful when looking at data sets to determine which student generally performs better or worse in a given time frame. However, while the mean score gives us a snapshot of performance, it doesn't show the entire picture regarding variability and consistency.

Standard Deviation

Standard deviation is an important concept used to measure the amount of variation or dispersion from the mean. It is represented by the symbol \( \sigma \).
A low standard deviation means that the data points are close to the mean, indicating consistency in performance.
Conversely, a high standard deviation indicates that the data points are spread out over a wider range of values.
This means more variability and less consistency. In our scenario, Dheeraja has the largest standard deviation of 11.25, suggesting her performance varies the most between assessments.

Data Analysis

Data analysis is the process of inspecting and interpreting data to identify trends and patterns. In the example given, we analyze data to make meaningful conclusions about student performance.
By looking at both mean scores and standard deviations, we can draw insights on who generally performs better and who exhibits consistent or inconsistent performance.
This two-dimensional analysis gives a more comprehensive view compared to simply assessing mean scores alone.
Observing standard deviation alongside mean allows us to see not only how well each student is performing on average but also how stable their performance is across different assessments.

Consistency in Data

Consistency in data refers to the similarity of values within a data set. A consistent set of data points will have a smaller standard deviation.
This is because the values tend to be closer to the mean, showing less variability.
In terms of the students' performance, consistency would mean that a student’s scores do not vary much from one assessment to the next.
This typically indicates a stable performance. In our example, Akshitha shows relatively high consistency with a lower standard deviation of 5.28, meaning her performance didn't fluctuate greatly between each assessment.