Comprehensive Summary of Descriptive Statistics, Quartiles, and Outliers

Measures of Central Tendency: Mode, Mean, and Median

Dataset Under Analysis: $4, 5, 7, 8, 9, 9, 10, 15, 18$
Mode:
- Definition: The mode is defined as the number that occurs most frequently within a dataset.
- Dataset Value: In the provided dataset, the mode is $9$ .
Mean:
- Definition: The mean represents the arithmetic average of a dataset.
- Calculation Method: Add all individual data values together and divide the total sum by the number of data items present.
- Academic Grade Example: If a student scores $70\%$ in English, $82\%$ in Maths, and $64\%$ in Geography across three subjects, the average is calculated by summing the three scores and dividing by $3$ : $\text{Average} = \frac{70\% + 82\% + 64\%}{3} = 72\%$
- Dataset Calculation:
  - Sum of all values: $4 + 5 + 7 + 8 + 9 + 9 + 10 + 15 + 18 = 85$
  - Total count of digits ( $n$ ): $9$
  - Mean formula application: $\text{Mean} = \frac{85}{9} = 9.44$
Median:
- Definition: The median is the value located at the exact halfway position (middle) of a dataset.
- Prerequisite Condition: Data values must be arranged in ascending order from smallest to largest before determining the median ( $4, 5, 7, 8, 9, 9, 10, 15, 18$ ).
- Counting Method: Counting inward step-by-step from both ends reveals that the middle value remaining is $9$ .
- Mathematical Position Formula: $\text{Position} = \frac{n + 1}{2}$
- Dataset Application:
  - With $n = 9$ item counts: $\text{Position} = \frac{9 + 1}{2} = \frac{10}{2} = 5$
  - Navigating to position $5$ starting from the first term gives the value $9$

Quartiles and Interquartile Calculations

Quartiles Overview: Quartiles divide an ordered dataset into four equal parts.
Lower Quartile ( $Q_1$ ):
- Definition: Represents the threshold for the lower quarter of the ordered data.
- Manual/Visual Method:
  - Ignore the central median ( $9$ ) and examine the remaining left-hand subset: $4, 5, 7, 8$
  - The halfway point between the inner values $5$ and $7$ yields $6$
- Mathematical Position Formula: $\text{Position} = \frac{n + 1}{4}$
- Dataset Application:
  - With $n = 9$ : $\text{Position} = \frac{9 + 1}{4} = \frac{10}{4} = 2.5$
  - Position $2.5$ is located exactly halfway between position $2$ (value $5$ ) and position $3$ (value $7$ ): $\text{Value} = \frac{5 + 7}{2} = 6$
Upper Quartile ( $Q_3$ ):
- Definition: Represents the threshold for the upper quarter (third quarter) of the ordered data.
- Mathematical Position Formula: $\text{Position} = 3 \times \left(\frac{n + 1}{4}\right)$
- Dataset Application:
  - With $n = 9$ : $\text{Position} = 3 \times 2.5 = 7.5$
  - Position $7.5$ lies halfway between position $7$ (value $10$ ) and position $8$ (value $15$ ): $\text{Value} = \frac{10 + 15}{2} = 12.5$

Standard Deviation and Calculator Procedure

Standard Deviation Overview: A measure of data spread calculated using a scientific calculator (such as a Casio calculator).
Step-by-Step Casio Calculator Procedure:
1. Press the MODE button.
2. Select option 3 for STAT mode.
3. Press option 1 (single-variable variance mode).
4. Input dataset values sequentially ( $4$ , $5$ , $7$ , $8$ , $9$ , $9$ , $10$ , $15$ , $18$ ), pressing the = key after entering each individual value.
5. Press the AC button to clear the display screen while keeping statistical data stored.
6. Press SHIFT, then press the 1 key (labelled STAT).
7. Select option 4 for VAR (variance).
8. Select option 3 (symbolized by standard deviation notation).
9. Press = to display the computed standard deviation.
Calculated Output: The standard deviation for the given dataset is $4.25$ when rounded to two decimal places.

Values Within Standard Deviation Boundaries

Determining Boundaries Around the Mean:
- Mean (average): $9.44$
- Standard Deviation: $4.25$
- Upper Boundary ( $\text{Mean} + 1\text{SD}$ ): $9.44 + 4.25 = 13.69$
- Lower Boundary ( $\text{Mean} - 1\text{SD}$ ): $9.44 - 4.25 = 5.19$
- Target Range: $[5.19, 13.69]$
Dataset Evaluation Against Boundaries:
- $4$ : Excluded (less than $5.19$ )
- $5$ : Excluded (less than $5.19$ )
- $7$ : Included
- $8$ : Included
- $9$ : Included
- $9$ : Included
- $10$ : Included
- $15$ : Excluded (greater than $13.69$ )
- $18$ : Excluded (greater than $13.69$ )
Result: Exactly $5$ numbers ( $7, 8, 9, 9, 10$ ) lie within one standard deviation of the mean.
Extension to Two Standard Deviations: To find values within two standard deviations, add an additional $4.25$ to the upper boundary ( $13.69 + 4.25$ ) and subtract $4.25$ from the lower boundary ( $5.19 - 4.25$ ) to establish a wider range.

Outlier Identification and Mathematical Criteria

Definition of an Outlier: A data point that is significantly too small or too large relative to the rest of the dataset, failing to match the statistical trend.
Impact of Outliers Example:
- Consider a classroom where four students obtain marks of $57\%$ , $61\%$ , $64\%$ , and $65\%$ , while a fifth student scores $99\%$ .
- Including the $99\%$ score drags the class average up to approximately $75\%$ . This creates an inaccurate impression of overall performance since most students scored in the high $50$ s to low $60$ s.
- Similarly, an isolated low score of $8\%$ would pull the overall average down dramatically.
- Outliers are routinely ignored or isolated because they skew statistical summaries away from accurate representations of data.
Interquartile Range (IQR):
- Formula: $\text{IQR} = Q_3 - Q_1$
- Dataset Calculation: $\text{IQR} = 12.5 - 6 = 6.5$
Lower Outlier Boundary Criteria:
- Formula: $\text{Lower Outlier Limit} = Q_1 - 1.5 \times \text{IQR}$
- Dataset Calculation: $\text{Lower Outlier Limit} = 6 - 1.5 \times 6.5 = 6 - 9.75 = -3.75$
- Rule: Any number lower than $-3.75$ is classified as a lower outlier.
- Evaluation: The dataset's minimum value is $4$ , which is greater than $-3.75$ . Therefore, there are no lower outliers.
Upper Outlier Boundary Criteria:
- Formula: $\text{Upper Outlier Limit} = Q_3 + 1.5 \times \text{IQR}$
- Dataset Calculation: $\text{Upper Outlier Limit} = 12.5 + 1.5 \times 6.5 = 12.5 + 9.75 = 22.25$
- Rule: Any number higher than $22.25$ is classified as an upper outlier.
- Evaluation: The dataset's maximum value is $18$ , which is less than $22.25$ . Therefore, there are no upper outliers.
Conclusion: All values in the dataset are clustered close enough together that no outliers exist.

Five-Number Summary

Definition: A standardized five-value statistical overview summarizing dataset distribution.
Components:
1. Minimum value ( $\text{Min}$ )
2. Lower Quartile ( $Q_1$ )
3. Median ( $\text{Med}$ )
4. Upper Quartile ( $Q_3$ )
5. Maximum value ( $\text{Max}$ )
Dataset Summary Representation:
- Minimum: $4$
- Lower Quartile ( $Q_1$ ): $6$
- Median: $9$
- Upper Quartile ( $Q_3$ ): $12.5$
- Maximum: $18$
- Bracketed format: $(4, 6, 9, 12.5, 18)$

Box and Whisker Diagram Construction and Skewness

Box and Whisker Diagram Concept: A visual representation of the Five-Number Summary plotted along a scaled axis or number line using a ruler.
Components:
- Box: Central section spanning from the Lower Quartile ( $Q_1$ ) to the Upper Quartile ( $Q_3$ ), split by the median line.
- Whiskers: Outer lines extending from the box to the Minimum and Maximum values.
Ruler Measurement Example (Limited to $14\,\text{cm}$ total space):
- Mark the Minimum ( $4$ ) at the starting point.
- Measure $2\,\text{cm}$ right to plot the Lower Quartile ( $6$ ).
- Measure $3\,\text{cm}$ further right to plot the Median ( $9$ ).
- Measure $3.5\,\text{cm}$ further right to plot the Upper Quartile ( $12.5$ ).
- Measure $5.5\,\text{cm}$ further right to plot the Maximum ( $18$ ) at the $14\,\text{cm}$ endpoint.
- Note: If using a full $30\,\text{cm}$ ruler with sufficient space, point positions can be placed directly on ruler markings matching the data values.
Evaluating Skewness:
- Skewness is determined by comparing the relative lengths of the left and right whiskers.
- Because the right-hand whisker is longer than the left-hand whisker, the dataset distribution is skewed to the right (right-skewed).