Statistics Basics: Data, Types, Presentation & Analysis

TAKEAWAYS:

• Data types (primary vs. secondary, qualitative vs. quantitative) determine how you collect and analyze information.
• Tabulation organizes raw data into meaningful tables (frequency distributions, cumulative frequency) for easier analysis.
• Graphical tools (bar charts, pie charts, histograms) visually reveal patterns in data, aiding decision-making.
• Measures of central tendency (mean, median, mode) summarize data distributions, while dispersion (range, quartiles) shows variability.
• Real-world applications span from eSewa transaction analysis to Daraz sales forecasting—statistics drives business insights.
• Exam focus: Master tabulation, graphical interpretation, and five-number summaries (min, Q1, median, Q3, max) for box plots.

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1. What is Statistics?

Statistics is the science of collecting, organizing, analyzing, interpreting, and presenting data to make informed decisions. It helps businesses understand trends, risks, and opportunities.

Why Study Statistics in Business?

• Decision-making: Banks use loan default rates (mean/median) to approve loans.
• Quality control: Factories (e.g., cement plants) track defect rates via histograms.
• Market research: Companies like Daraz analyze customer purchase patterns (frequency distributions).

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2. Types of Data

Data can be classified based on source and nature:

A. By Source

B. By Nature

Example:

• Primary Data: eSewa collects user transaction amounts (quantitative, continuous) via its app.
• Secondary Data: NTC uses past electricity consumption data (quantitative, discrete) to plan infrastructure.

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3. Data Collection Methods

Worked Example: Suppose Pathao wants to study rider wait times at pick-up points. They collect data for 100 rides:

• Primary Data: Timestamps from Pathao’s app (continuous, quantitative).
• Secondary Data: Traffic reports from Kathmandu Metropolitan City (categorical, qualitative).

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4. Tabulation: Organizing Data

Raw data is meaningless without organization. Frequency distributions and tables simplify analysis.

A. Ungrouped Frequency Distribution

For discrete data (e.g., number of customers per hour at a Khalti kiosk):

Key Terms:

• Class Interval: Groups for continuous data (e.g., "10–20 customers").
• Class Mark: Midpoint of a class (e.g., for 10–20, mark = 15).
• Cumulative Frequency (CF): Running total of frequencies.

B. Grouped Frequency Distribution

For continuous data (e.g., ages of TU students):

Worked Example: Given the following data on monthly salaries (in Rs. '000) of 20 employees at a Daraz warehouse: 50, 45, 60, 70, 55, 65, 40, 80, 75, 50, 60, 55, 45, 70, 85, 65, 50, 75, 40, 60

Step 1: Choose class intervals (e.g., 40–50, 50–60, etc.). Step 2: Tally frequencies:

Step 3: Calculate relative frequency (percentage):

• For 50–60: $\frac{6}{20}\times 100=30\%$.

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5. Graphical Presentation of Data

Visuals make data intuitive. Common graphs:

A. Bar Chart

• Use: Compare discrete categories (e.g., sales by product).
• Example: Monthly sales of Daraz’s electronics vs. groceries.

B. Pie Chart

• Use: Show proportions (e.g., market share of Ncell vs. NTC).
• Rule: Total = 100%.

C. Histogram

• Use: Show distribution of continuous data (e.g., heights of TU students).
• Key: No gaps between bars.

D. Frequency Polygon

• Use: Smooth curve for trends (e.g., Khalti transaction growth over months).

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6. Measures of Central Tendency

These summarize data with a single value.

A. Mean (Average)

$$\text{Mean}=\frac{\sum X}{N}$$

• Use: Overall trend (e.g., average loan amount at NMB Bank).
• Limitation: Affected by outliers (e.g., a billionaire skews average income).

Worked Example: Calculate the mean age of 10 TU students: 20, 22, 21, 23, 20, 25, 19, 24, 22, 21. $$\text{Mean}=\frac{20+22+21+23+20+25+19+24+22+21}{10}=\frac{217}{10}=21.7\text{ years}$$

B. Median

• Use: Middle value (robust to outliers).
• Steps:
  1. Arrange data in order.
  2. Find the middle value (or average of two middle values).

Worked Example: For the same age data (ordered): 19, 20, 20, 21, 21, 22, 22, 23, 24, 25.

• Median = average of 5th and 6th values = $\frac{21+22}{2}=21.5$.

C. Mode

• Use: Most frequent value (e.g., most common order size on Daraz).
• Limitation: Data may have no mode or multiple modes.

Worked Example: *In the age data, 20, 21, and 22 each appear twice → bimodal.

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7. Measures of Dispersion

Show how spread out data is.

A. Range

$$\text{Range}=\text{Max}-\text{Min}$$

• Example: Salaries at a startup: 40,000–100,000 → Range = 60,000.

B. Quartiles and Five-Number Summary

Used for box plots (critical for TU exams!).

Steps:

1. Order data: $x_1,x_2,\dots ,x_n$.
2. Find:
   • Q1 (1st Quartile): Median of the lower half.
   • Q3 (3rd Quartile): Median of the upper half.
   • Median (Q2): Already calculated.
   • Min/Max: Smallest/largest values.

Worked Example: Given data: 39, 26, 15, 8, 70, 11, 45, 60, 20, 32, 52. Step 1: Order → 8, 11, 15, 20, 26, 32, 39, 45, 52, 60, 70. Step 2: Five-number summary:

• Min = 8
• Q1 = Median of {8, 11, 15, 20, 26} = 15
• Median (Q2) = 32
• Q3 = Median of {39, 45, 52, 60, 70} = 52
• Max = 70

Box Plot Interpretation:

• Skewness:
  • If median > mean → left-skewed (long tail on left).
  • If median < mean → right-skewed (long tail on right).
  • Here, median (32) < mean (~35) → right-skewed.

C. Standard Deviation (σ)

Measures average deviation from the mean. $$\sigma =\sqrt{\frac{\sum (X-\mu {)}^2}{N}}$$ Worked Example: Given mean (μ) = 20, CV = 30%, calculate σ. $$\text{CV}=\frac{\sigma }{\mu }\times 100⟹30=\frac{\sigma }{20}\times 100⟹\sigma =6$$

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In the Real World

1. eSewa Transaction Analysis

   • Idea Used: Frequency Distribution + Mean/Median
   • How: eSewa categorizes transactions by amount (e.g., 100–500, 500–1000) to identify peak spending ranges. The median transaction value helps set default payment limits.

2. Daraz Sales Forecasting

   • Idea Used: Time Series + Histogram
   • How: Daraz plots monthly sales (frequency polygon) to spot trends (e.g., spikes during Dashain). The standard deviation of daily orders helps stock inventory.

3. NMB Bank Loan Approvals

   • Idea Used: Five-Number Summary + Box Plots
   • How: Bank analysts use the interquartile range (IQR = Q3 – Q1) to flag high-risk borrowers (e.g., applicants with incomes outside Q1–Q3). A right-skewed distribution of loan amounts suggests most borrowers take small loans, but a few take large ones.

4. NTC Electricity Demand Planning

   • Idea Used: Grouped Frequency Distribution
   • How: NTC groups hourly electricity usage into classes (e.g., 0–50 MW, 50–100 MW) to predict peak demand. The mode reveals the most common usage range.

5. Pathao Driver Earnings

   • Idea Used: Mean vs. Median
   • How: Pathao’s earnings data is right-skewed (a few drivers earn very high tips). The median (not mean) gives a fairer picture of "typical" earnings.

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Exam Tip

1. Tabulation is Key:

   • Always organize raw data into frequency tables before plotting.
   • For grouped data, calculate class marks and relative frequencies.

2. Graphs Must Be Labeled:

   • Title, axes, and units are mandatory. A pie chart without percentages is incomplete.

3. Five-Number Summary > Mean Alone:

   • TU exams often ask for box plots. Always calculate min, Q1, median, Q3, max and comment on skewness.

4. Real-World Applications:

   • Link answers to business scenarios (e.g., "Like Daraz, a company can use histograms to analyze customer purchase patterns").
   • For standard deviation, relate to risk (e.g., "High σ in loan defaults means higher risk for banks").

5. Common Pitfalls:

   • Bar charts vs. histograms: Bars in histograms touch; bars in bar charts don’t.
   • Median position: For n data points, median is at $\frac{n+1}{2}$. For n=11, it’s the 6th value.

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Practice Questions (TU-Style)

1. Tabulation: Given the following data on monthly internet usage (GB) of 20 students: 5, 8, 6, 10, 7, 9, 12, 4, 6, 8, 10, 11, 5, 7, 9, 12, 15, 6, 8, 10.

   • Prepare a grouped frequency distribution with classes 4–6, 6–8, etc.
   • Draw a frequency polygon.

2. Graphical Interpretation: The following table shows the number of accidents at a Kathmandu intersection by hour:

   Hour	8 AM	9 AM	10 AM	11 AM	12 PM
   Accidents	5	8	12	6	4

   • Draw a bar chart and comment on the trend.

3. Five-Number Summary: For the data: 12, 15, 14, 10, 8, 6, 5, 18, 20, 16, 14, 12.

   • Calculate the five-number summary.
   • Is the distribution skewed? Justify.

4. Standard Deviation: The mean height of 100 TU students is 165 cm with a CV of 2.5%. Calculate the standard deviation.