Grade 10 MATH Term 1 Week 8 to 9: Calculating Measures of Positions | MATATAG 2nd Quarter-Tagalog

Main ideas / lesson goals

The lesson explains measures of position for ungrouped data, focusing on how to compute:

• Quartiles (Q1, Q2, Q3)
• Interquartile Range (IQR)
• Outliers
• Deciles (D1 to D9) (with examples + interpretation)
• Percentiles (P1 to P99) (with examples + interpretation)

Key concept: Measures of position tell where a value lies in an ordered dataset—often describing the percentage of observations that are at/below or at/above that measure.

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Methodology / formulas taught (step-by-step)

A) General process for quartiles/deciles/percentiles (ungrouped data)

1. Arrange the data in ascending order (least to greatest).
2. Compute the position of the desired quartile/decile/percentile using the appropriate formula.
3. If the computed position is not an integer, compute the exact value using linear interpolation between the two surrounding data points.
4. Use the computed values for further tasks (IQR, outliers) or for interpretation.

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B) Quartiles (Q1, Q2, Q3) using the n + 1 position formula + linear interpolation

Quartile position formula

• Position of Qk = k(n + 1) / 4
  • where k = 1, 2, 3
  • n = number of observations

Meaning

• Q1 (lower quartile): about 25% of data are at or below Q1
• Q2 (median / middle quartile): about 50% are at or below Q2
• Q3 (upper quartile): about 75% are at or below Q3

Linear interpolation (when position is fractional)

If the position equals j + d where:

• j = smaller (lower) position index
• d = decimal part (fraction)
• value at j = smaller number
• value at j+1 = larger number

Interpolation steps:

1. Compute difference = (larger value − smaller value)
2. Multiply by decimal part = difference × d
3. Add to smaller value = smaller value + (difference × d)

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C) Interquartile Range (IQR)

• IQR = Q3 − Q1

Interpretation: The IQR measures the spread of the middle 50% of the data.

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D) Outliers (using fences)

Outlier definition: a value unusually low or high compared to the rest (extreme values).

Fence rule

• Lower fence = Q1 − 1.5(IQR)
• Upper fence = Q3 + 1.5(IQR)

Decision rule

• If a value is < lower fence or > upper fence, it is an outlier
• Otherwise, no outliers are present.

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E) Deciles (Dk, where k = 1 to 9) + interpretation

Decile position formula

• Position of Dk = k(n + 1) / 10
  • where k = 1, 2, …, 9
  • n = number of observations

Linear interpolation is used if the position is fractional (same approach as quartiles).

Interpretation taught

• Dk means approximately k/10 (×100%) of the students have a score at or below Dk.
• The remaining portion has a score at or above Dk.

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F) Percentiles (Pk, where k = 1 to 99) + interpretation

Percentile position formula

• Position of Pk = k(n + 1) / 100
  • where k = 1, 2, …, 99
  • n = number of observations

Linear interpolation is used if the position is fractional.

Interpretation taught

• Pk means approximately k% of values are at or below Pk.
• Approximately (100 − k)% are at or above Pk.

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Examples covered (what was computed and what conclusions were drawn)

Example 1: Quartiles + IQR + Outliers (11 quiz scores)

Computed:

• Q1 = 56
• Q2 = 68
• Q3 = 80
• IQR = Q3 − Q1 = 24

Outlier fences:

• Lower fence = 20
• Upper fence = 116

Conclusion: All scores are between the fences → no outliers.

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Example 2: Quartiles + IQR + Outliers (10 test scores)

Computed (with linear interpolation):

• Q1 = 61.5
• Q2 = 67
• Q3 = 72.5
• IQR = 72.5 − 61.5 = 11

Outlier fences:

• Lower fence = 45
• Upper fence = 89

Conclusion: A score 10 is below the lower fence → 10 is an outlier.

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Example 3: 4th Decile (14 math test scores)

Computed:

• D4 = 74

Interpretation:

• Approximately 40% scored 74 or lower
• Approximately 60% scored 74 or higher

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Example 4: 6th Decile (11 completion times)

Computed:

• D6 = 31

Interpretation:

• Approximately 60% completed in 31 minutes or less
• Approximately 40% took 31 minutes or more

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Example 5: 35th Percentile (19 science test scores)

Computed:

• P35 = 68

Interpretation:

• Approximately 35% scored 68 or lower
• Approximately 65% scored 68 or higher

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Example 6: 60th Percentile (15 seedling heights)

Computed (with linear interpolation):

• P60 = 26.8 cm

Interpretation:

• Approximately 60% have heights 26.8 cm or less
• Approximately 40% have heights 26.8 cm or more

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Wrap-up / key takeaways

• Always arrange data first.
• Use:
  • Quartiles: position = k(n+1)/4
  • Deciles: position = k(n+1)/10
  • Percentiles: position = k(n+1)/100
• When the position is fractional, apply linear interpolation.
• For quartiles:
  • IQR = Q3 − Q1
  • Outliers determined by fences using ± 1.5(IQR).

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Speakers / sources featured

• Math Easy YouTube Channel / Presenter (unnamed; the instructor delivering the Tagalog/English lesson)
• MATATAG 2nd Quarter (referenced in the video title)