Descriptive Statistics

Given: Two types of graphical representations — Bar Graph and Histogram.

Differences between Bar Graph and Histogram:

| Feature | Bar Graph | Histogram |
|---|---|---|
| Type of data | Categorical (discrete) data | Continuous (grouped) data |
| Bars | Bars are separated by gaps | Bars are adjacent (no gaps) |
| X-axis | Represents categories or discrete values | Represents class intervals (continuous) |
| Width of bars | All bars have equal width; width has no meaning | Width represents the class interval size; width is meaningful |
| Rearrangement | Bars can be rearranged in any order | Bars cannot be rearranged; order is fixed |
| Purpose | Compares different categories | Shows frequency distribution of continuous data |
| Base | Each bar stands on its own base | Bars share a common continuous base |

Conclusion: A bar graph is used for discrete/categorical data where gaps between bars indicate distinct categories, while a histogram is used for continuous frequency distributions where the area of each bar represents the frequency of that class interval.

Given: Various types of data sets.

Concept: Different types of graphs are suitable for different types of data.

Discussion of suitable graph types:

| Type of Data | Most Suitable Graph | Reason |
|---|---|---|
| Categorical comparison (e.g., population of different states) | Bar Graph | Compares discrete categories clearly |
| Parts of a whole (e.g., budget allocation) | Pie Chart | Shows proportion/percentage of each part |
| Continuous frequency distribution (e.g., marks of students) | Histogram / Frequency Polygon | Represents class intervals and frequencies |
| Trend over time (e.g., GDP growth over years) | Line Graph | Shows change/trend over a continuous period |
| Two variables relationship (e.g., height vs weight) | Scatter Plot | Shows correlation between two variables |

Can data be represented in two or more ways?

Yes, it is possible to represent the same data graphically in two or more ways. For example:
- A continuous frequency distribution can be shown using a Histogram, a Frequency Polygon, or an Ogive (cumulative frequency curve).
- Categorical data can be shown using both a Bar Graph and a Pie Chart.

Conclusion: While multiple representations are possible, the choice of graph should be guided by the nature of the data and the purpose of analysis. The most appropriate graph makes the data easier to interpret and compare.

Given: Tax rate data for different income slabs under current and new tax regimes.

Most Suitable Graph: Double/Grouped Bar Graph

Justification: The data involves comparison of two sets of values (Current Tax Rates vs New Tax Rates) across different income slabs (categories). A grouped (double) bar graph is most suitable because:
- It allows direct visual comparison between two data series side by side.
- The income slabs are discrete categories, making a bar graph appropriate.
- It clearly shows the reduction in tax rates across slabs.

Steps to draw the Grouped Bar Graph:

Step 1: Mark the income slabs on the X-axis.

Step 2: Mark the tax rates (in %) on the Y-axis.

Step 3: For each income slab, draw two bars side by side — one for the Current Tax Rate (e.g., in blue) and one for the New Tax Rate (e.g., in orange).

Step 4: Add a legend to distinguish the two bars.

Data for plotting:

| Income Slab | Current Rate (%) | New Rate (%) |
|---|---|---|
| 0–2.5 L | 0 | 0 |
| 2.5–5 L | 5 | 0 |
| 5–7.5 L | 20 | 10 |
| 7.5–10 L | 20 | 15 |
| 10–12.5 L | 30 | 20 |
| 12.5–15 L | 30 | 25 |
| Above 15 L | 30 | 30 |

Observation from the graph: The new tax regime offers lower tax rates for all income slabs up to Rs. 15 Lakh, while the rate remains the same (30%) for income above Rs. 15 Lakh. This shows the government's intent to reduce the tax burden on middle-income groups.

Given: Fiscal Deficit and Revenue Deficit data (as % of GDP) over five years/periods.

Most Suitable Graph: Double Line Graph (Multiple Line Graph)

Justification: The data shows the trend of two types of deficits over a time period (from 2018-19 to 2022-23). A line graph is most suitable because:
- It effectively shows trends and changes over time.
- Two lines can be drawn simultaneously to compare Fiscal Deficit and Revenue Deficit.
- It clearly shows whether deficits are increasing or decreasing over the years.

Steps to draw the Double Line Graph:

Step 1: Mark the years/periods on the X-axis: 2018-19, 2019-20, 2020-21, 2021-22, 2022-23.

Step 2: Mark the deficit percentages (0% to 4%) on the Y-axis.

Step 3: Plot the points for Fiscal Deficit: (2018-19, 3.4), (2019-20, 3.8), (2020-21, 3.5), (2021-22, 3.3), (2022-23, 3.1) and join them with a line.

Step 4: Plot the points for Revenue Deficit: (2018-19, 2.4), (2019-20, 2.4), (2020-21, 2.7), (2021-22, 2.3), (2022-23, 1.9) and join them with a different coloured line.

Step 5: Add a legend to distinguish the two lines.

Observation: Both Fiscal Deficit and Revenue Deficit show a peak in 2019-20 and 2020-21 respectively, after which they are targeted to decline. The Fiscal Deficit is consistently higher than the Revenue Deficit throughout the period. The government aims to achieve fiscal consolidation by 2022-23.

Given: Data: $3, 6, 2, 1, 7, 5$; $n = 6$

Formula:
$$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{n}(x_i - \bar{x})^2}$$

Step 1: Find the Mean
$$\bar{x} = \frac{3 + 6 + 2 + 1 + 7 + 5}{6} = \frac{24}{6} = 4$$

Step 2: Find $(x_i - \bar{x})$ and $(x_i - \bar{x})^2$ for each value

| $x_i$ | $x_i - \bar{x}$ | $(x_i - \bar{x})^2$ |
|---|---|---|
| 3 | $3 - 4 = -1$ | 1 |
| 6 | $6 - 4 = 2$ | 4 |
| 2 | $2 - 4 = -2$ | 4 |
| 1 | $1 - 4 = -3$ | 9 |
| 7 | $7 - 4 = 3$ | 9 |
| 5 | $5 - 4 = 1$ | 1 |
| Total | | 28 |

Step 3: Calculate Standard Deviation
$$\sigma = \sqrt{\frac{\sum(x_i - \bar{x})^2}{N}} = \sqrt{\frac{28}{6}} = \sqrt{4.\overline{6}} = \sqrt{\frac{14}{3}}$$

$$\sigma = \sqrt{4.667} \approx 2.16$$

Therefore, the Standard Deviation $\sigma \approx 2.16$

Given: Data: $40, 38, 42, 60, 72, 54$; $n = 6$

Formula:
$$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{n}(x_i - \bar{x})^2}$$

Step 1: Find the Mean
$$\bar{x} = \frac{40 + 38 + 42 + 60 + 72 + 54}{6} = \frac{306}{6} = 51$$

Step 2: Find $(x_i - \bar{x})$ and $(x_i - \bar{x})^2$

| $x_i$ | $x_i - \bar{x}$ | $(x_i - \bar{x})^2$ |
|---|---|---|
| 40 | $40 - 51 = -11$ | 121 |
| 38 | $38 - 51 = -13$ | 169 |
| 42 | $42 - 51 = -9$ | 81 |
| 60 | $60 - 51 = 9$ | 81 |
| 72 | $72 - 51 = 21$ | 441 |
| 54 | $54 - 51 = 3$ | 9 |
| Total | | 902 |

Step 3: Calculate Standard Deviation
$$\sigma = \sqrt{\frac{902}{6}} = \sqrt{150.33} \approx 12.26$$

Therefore, the Standard Deviation $\sigma \approx 12.26$

Given:
- Set 1: $3, 6, 2, 1, 7, 5$
- Set 2 (obtained by $y = 2x + 5$): $11, 17, 9, 7, 19, 15$

From Question 5, for Set 1:
$$\bar{x}_1 = 4, \quad \sigma_1 \approx 2.16$$

For Set 2 — Step 1: Find the Mean
$$\bar{x}_2 = \frac{11 + 17 + 9 + 7 + 19 + 15}{6} = \frac{78}{6} = 13$$

Step 2: Find Standard Deviation of Set 2

| $y_i$ | $y_i - \bar{y}$ | $(y_i - \bar{y})^2$ |
|---|---|---|
| 11 | $-2$ | 4 |
| 17 | $4$ | 16 |
| 9 | $-4$ | 16 |
| 7 | $-6$ | 36 |
| 19 | $6$ | 36 |
| 15 | $2$ | 4 |
| Total | | 112 |

$$\sigma_2 = \sqrt{\frac{112}{6}} = \sqrt{18.\overline{6}} = \sqrt{\frac{56}{3}} \approx 4.32$$

Step 3: Establish the Relationship

The transformation used is $y = 2x + 5$.

Relationship between Means:
$$\bar{y} = 2\bar{x} + 5 \implies 13 = 2(4) + 5 = 13 \checkmark$$

Relationship between Standard Deviations:
$$\sigma_y = |2| \times \sigma_x \implies \sigma_2 = 2 \times \sigma_1$$
$$4.32 = 2 \times 2.16 \checkmark$$

Conclusion:
- The mean of Set 2 = $2 \times$ (mean of Set 1) $+ 5$, i.e., $\bar{y} = 2\bar{x} + 5$.
- The standard deviation of Set 2 = $2 \times$ (standard deviation of Set 1), i.e., $\sigma_y = 2\sigma_x$.
- Key Principle: When each value is multiplied by a constant $k$ and a constant $c$ is added (i.e., $y = kx + c$), the mean transforms as $\bar{y} = k\bar{x} + c$, but the standard deviation transforms as $\sigma_y = |k|\sigma_x$ (the additive constant $c$ does not affect the standard deviation).

Given: Two statistical measures — Percentile and Percentile Rank.

Differences between Percentile and Percentile Rank:

| Feature | Percentile | Percentile Rank |
|---|---|---|
| Definition | A value below which a given percentage of observations fall | The percentage of scores in a distribution that fall at or below a given score |
| What it gives | A score/value corresponding to a given percentage | A percentage/rank corresponding to a given score |
| Direction | Percentage → Score | Score → Percentage |
| Example | $P_{75}$ = 80 means 75% of students scored below 80 | PR of score 80 = 75 means the student scored better than 75% of students |
| Use | Used to find the cut-off score for a given percentage | Used to find the relative standing of a particular score |
| Formula | $P_k = l + \left(\dfrac{\frac{kN}{100} - cf}{f}\right) \times h$ | $PR = \dfrac{100}{N}\left(cf + \dfrac{x}{h} \times f\right)$ |

In summary: Percentile converts a percentage into a score, while Percentile Rank converts a score into a percentage. They are inverse operations of each other.

Note: Example 43 refers to the grouped data table provided in the solution section of the OCR text. The data table is:

| Class Interval | f | cf | Percentage of cf |
|---|---|---|---|
| 93–97 | 4 | 59 | 100 |
| 88–92 | 7 | 55 | 93.22 |
| 83–87 | 5 | 48 | 81.36 |
| 78–82 | 8 | 43 | 72.88 |
| 73–77 | 3 | 35 | 59.32 |
| 68–72 | 6 | 32 | 54.24 |
| 63–67 | 7 | 26 | 44.07 |
| 58–62 | 10 | 19 | 32.30 |
| 53–57 | 5 | 9 | 15.25 |
| 48–52 | 4 | 4 | 6.78 |

Here $N = 59$, class size $h = 5$.

Formula for Percentile Rank of grouped data:
$$PR = \frac{100}{N}\left(cf + \frac{x}{h} \times f\right)$$
where $x$ = score $-$ lower true boundary of the class, $cf$ = cumulative frequency below the class, $f$ = frequency of the class.

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Calculating $PR_{70}$ (Percentile Rank of score 70):

Score = 70 lies in class interval 68–72.

- Lower true boundary $l = 67.5$
- $x = 70 - 67.5 = 2.5$
- $f = 6$ (frequency of class 68–72)
- $cf = 26$ (cumulative frequency below class 68–72)
- $h = 5$, $N = 59$

$$PR_{70} = \frac{100}{59}\left(26 + \frac{2.5}{5} \times 6\right) = \frac{100}{59}\left(26 + 3\right) = \frac{100 \times 29}{59}$$

$$PR_{70} = \frac{2900}{59} \approx 49.15$$

$\therefore$ Percentile Rank of score 70 $\approx 49.15$

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Calculating $PR_{50}$ (Percentile Rank of score 50):

Score = 50 lies in class interval 48–52.

- Lower true boundary $l = 47.5$
- $x = 50 - 47.5 = 2.5$
- $f = 4$ (frequency of class 48–52)
- $cf = 0$ (no class below 48–52)
- $h = 5$, $N = 59$

$$PR_{50} = \frac{100}{59}\left(0 + \frac{2.5}{5} \times 4\right) = \frac{100}{59}\left(0 + 2\right) = \frac{200}{59}$$

$$PR_{50} = \frac{200}{59} \approx 3.39$$

$\therefore$ Percentile Rank of score 50 $\approx 3.39$

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Interpretation:
- A student scoring 70 has performed better than approximately 49.15% of the students.
- A student scoring 50 has performed better than approximately 3.39% of the students.