STATGEN CLASS 2023 #6 Metode Statistika 1 (SATS4121) Pertemuan 3

Main ideas / lessons conveyed

1. Purpose of statistics in descriptive form

   • Descriptive statistics summarizes data so that readers get useful information.
   • Data presentation is covered first (frequency tables, bar charts, histograms, pie charts, and box plots/box-and-whisker plots), then moves to numerical measures.

2. Data presentations (review)

   • Frequency distribution tables organize values by frequency.
   • Bar charts and histograms visualize frequency across categories/intervals.
   • Pie charts show proportional parts of the whole.
   • Box plot (box diagram) summarizes:
     • Median (center)
     • Quartiles: splits data into 4 equal parts (each 25%)
     • Whiskers: extend to the smallest/largest (or extremes as shown)
     • Outliers: values far from the majority (described as possible “extreme” points)

3. Numerical measures (centralization and dispersion)

   • Numerical measures are divided into:
     • Measures of centralization (location): tell where data tends to be.
     • Measures of dispersion (spread/distribution width): tell how spread out the data is.

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Detailed methodology / instructions included

A) Box plot interpretation (quartiles and outliers)

• Identify:

  • Whisker ends: smallest and largest shown values
  • Median: center line in the box
  • Lower quartile (Q1): lower edge of the box
  • Upper quartile (Q3): upper edge of the box

• Interpret spread from the box plot:

  • The box shows the interquartile range; each quartile represents 25% of the data.
  • If the box/median is shifted upward, larger values dominate.
  • Points far away from the whiskers are outliers.

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B) Measures of centralization

1. Mean (average)

For ungrouped/single data

• Compute:
  • Add all values
  • Divide by number of data (n)

For grouped data

• Steps:
  • Determine class midpoints for each interval (middle of class boundaries).
  • Multiply each midpoint by its frequency: ( f \times x_{\text{mid}} )
  • Sum all ( f x ) values.
  • Divide by total frequency ( n = \sum f ).
• Key point: you can’t directly average raw values because data are given by intervals.

2. Median

For ungrouped/single data

• Steps:
  • Sort data from smallest to largest.
  • If total count is odd, the median is the middle value.
  • If total count is even, the median is the average of the two central values.

For grouped data

• Conceptual procedure:
  • Find the median class: the class containing position ( n/2 ).
  • Use:
    • Lower limit / boundary of median class
    • Class width (length)
    • Cumulative frequency before median class
    • Frequency inside median class
• Nuance:
  • Some references use lower limit, others use lower edge (lower limit minus 0.5).

3. Quartiles (Q1, Q2, Q3)

• Steps are similar to median but use these positions:
  • Q1 corresponds to ( n/4 )
  • Q2 (median) corresponds to ( n/2 )
  • Q3 corresponds to ( 3n/4 )

For grouped data

• Identify the relevant quartile class interval and use:
  • Class lower boundary/edge
  • Class width
  • Cumulative frequency before the quartile class
  • Frequency in that class

4. Mode

For grouped data

• Steps:
  • Identify mode class (the class with the largest frequency).
  • Determine:
    • ( D_1 ): mode class frequency − frequency of previous class
    • ( D_2 ): mode class frequency − frequency of next class
  • Plug into the grouped mode formula using:
    • Lower limit of mode class
    • Class width
    • ( D_1 ), ( D_2 )

5. Percentiles (and relation to deciles)

• Concept:
  • Median divides into 2 parts, quartiles into 4 parts, percentiles into 100 parts.
  • Deciles correspond to 10 parts (percentile 100% corresponds to the final part).
• For grouped data (described):
  • Identify the class containing the percentile position ( \frac{PR}{100} \times n ).
  • Use:
    • lower boundary/edge
    • class width
    • cumulative frequency before percentile class
    • frequency of percentile class
• Example logic:
  • To find P90, locate the class containing the ~(0.9n)-th data position using cumulative frequency.

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C) Measures of dispersion (spread)

1. Range

• Formula:
  • Range = maximum − minimum

2. Variance and standard deviation

Variance purpose

• Measures how far data values deviate/spread from the mean.

Sample variance vs population variance

• Key rule:
  • Sample variance uses divisor ( n - 1 )
  • Population variance uses divisor ( n )
• Why:
  • Sample is taken from a larger population; using ( n-1 ) helps avoid bias.

Formulas described (grouped vs single)

• Sample variance:
  • squared deviations from mean with divisor ( n-1 )
• Population variance:
  • squared deviations from mean with divisor ( n )
• For grouped data:
  • Use class midpoints as representative ( x )
  • Incorporate frequencies ( f ) in summations
  • Separate grouped forms exist for sample vs population variance

Standard deviation

• Steps:
  • Compute variance first.
  • Take square root of the variance:
    • ( s = \sqrt{s^2} ) (sample)
    • ( \sigma = \sqrt{\sigma^2} ) (population)

3. Coefficient of variation (CV)

• Meaning:
  • Relative variability: compares spread relative to the mean.
  • Useful for comparing datasets with different means/standard deviations.
• Formula described:
  • ( CV = \frac{s}{\bar{x}} \times 100\% ) (and similarly for population form depending on context)

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Practice / exercise component

• The instructor plans practice questions/exercises:
  • Given grouped data with frequencies, students calculate:
    • mean, mode, median
    • quartiles (notes that quartile 2 equals the median, so it may not need separate calculation)
    • variance
    • standard deviation
    • coefficient of variation
• Time suggested:
  • Around 5–10 minutes to attempt, then continue.

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Speakers / sources featured

• Indrio Murti (main instructor/resource person)
• YouTube video (source of auto-generated subtitles)