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Mathematics and Statistics · Class 11 Commerce

Ch 10Partition Values — Class 11 Mathematics and Statistics, concept-first.

The median of a set of data is the middle value: it splits an ordered data set into two equal halves, with as many observations below it as above it. Partition values extend exactly this idea — they are the values that divide an ordered data set into a chosen number of equal parts.

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Key concepts

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Partition Values for Ungrouped (Raw) Data

Arrange the observations in ascending order; then , , are the values at positions , , respectively. A non-integer position is resolved by linear interpolation between the neighbouring observations.

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Chapter contents

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Partition Values — The Idea of Position

The median of a set of data is the middle value: it splits an ordered data set into two equal halves, with as many observations below it as above it.

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Partition Values for Ungrouped (Raw) Data

For ungrouped (raw) data, first arrange the observations in ascending order. Each partition value is then the observation lying at a particular position in that ordered list.

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Partition Values for a Discrete Frequency Distribution

When data are given as distinct values with frequencies (a discrete frequency distribution), the observations are not listed one by one, so we locate a partition value through the less-than cumulative…

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Partition Values for a Continuous (Grouped) Frequency Distribution

For a continuous (class-interval) frequency distribution the exact observations are unknown — we know only how many fall in each class — so a partition value is found by interpolation inside the class…

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Graphical Location using the Ogive

Partition values of a continuous distribution can be read straight off an ogive — the graph of the less-than cumulative frequency curve.

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Interpretation and Relationships

A partition value only becomes useful once its meaning is stated in words.

Exercises

+Show 5 questions5 questions
  1. Q9Find $Q_1$, $Q_2$ and $Q_3$ for the data: $5, 8, 12, 15, 20, 25, 30$.Free
  2. Q10Find the third quartile $Q_3$ for the continuous distribution: classes $0\text{–}20, 20\text{–}40, 40\text{–}60, 60\text{–}80, 80\text{–}100…Free
  3. Q11Find the median (second quartile $Q_2$) of the discrete distribution: $x = 1, 2, 3, 4, 5$ with frequencies $f = 3, 5, 8, 6, 3$.Preview
  4. Q12For the data $10, 20, 30, 40, 50, 60, 70, 80, 90$, compute $Q_2$, $D_5$ and $P_{50}$ and hence verify that $Q_2 = D_5 = P_{50} = \text{Media…Preview
  5. Q13In a national scholarship examination taken by $2000$ candidates, the cut-off for an interview call is set at the $90$th percentile. Explain…Preview

More questions

+Show 8 questions8 questions
  1. Example 1Find the three quartiles $Q_1$, $Q_2$ and $Q_3$ for the data: $12, 18, 25, 30, 35, 40, 48, 52, 60$.Free
  2. Example 2For the marks $15, 20, 22, 25, 28, 30, 35, 40, 45, 50$ of $10$ students, find (i) the fourth decile $D_4$ and (ii) the $70$th percentile $P_…Free
  3. Example 3Find $Q_1$, $Q_2$ and $Q_3$ for the discrete frequency distribution: values $x = 10, 20, 30, 40, 50$ with frequencies $f = 4, 7, 12, 9, 3$.Free
  4. Example 4For the distribution $x = 10, 20, 30, 40, 50$ with $f = 4, 7, 12, 9, 3$, find (i) the seventh decile $D_7$ and (ii) the $25$th percentile $P…Preview
  5. Example 5Find $Q_1$, $Q_2$ and $Q_3$ for the continuous frequency distribution: classes $0\text{–}10, 10\text{–}20, 20\text{–}30, 30\text{–}40, 40\te…Preview
  6. Example 6For the distribution classes $0\text{–}10, 10\text{–}20, 20\text{–}30, 30\text{–}40, 40\text{–}50$ with frequencies $5, 8, 15, 12, 10$, find…Preview
  7. Example 7Using the ogive of the distribution in Worked Example 5 (classes $0\text{–}50$, frequencies $5, 8, 15, 12, 10$, $N = 50$), explain how to lo…Preview
  8. Example 8In an entrance test taken by many candidates, a student's score is reported to be at the $80$th percentile, $P_{80} = 68$ marks. Interpret t…Preview