Basic Statistics

Introduction to statistics

Quartiles in Statistics: Relative Measure of Observation

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Quartiles in Statistics

Like Percentiles and Deciles, Quartiles is a type of Quantile, which is a measure of the relative standing of observation within the data set. The Quartiles values are three points that divide the data into four equal parts each group comprising a quarter of the data (the first quartile $Q_1$, second quartile $Q_2$ (also median), and the third quartile $Q_3$) in the order statistics.

The first quartile, (also known as the lower quartile $Q_1$) is the value of order statistic that exceeds 1/4 of the observations and less than the remaining 3/4 observations. The third quartile known as the upper quartile is the value in the order statistic that exceeds 3/4 of the observations and is less than the remaining 1/4 observations, while the second quartile is the median.

Quartiles in Statistics for Ungrouped Data

For ungrouped data, the quartiles are calculated by splitting the order statistic at the median and then calculating the median of the two halves. If $n$ is odd, the median can be included on both sides.

Example: Find the $Q_1, Q_2$ and $Q_3$ for the following ungrouped data set 88.03, 94.50, 94.90, 95.05, 84.60.Solution: We split the order statistic at the median and calculated the median of two halves. Since $n$ is odd, we can include the median in both halves. The order statistic is 84.60, 88.03, 94.50, 94.90, 95.05.

Quartiles in Statistics: Relative Measure of Observation

\begin{align*}
Q_2&=median=Y_{(\frac{n+1}{2})}=Y_{(3)}\\
&=94.50  (\text{the third observation})\\
Q_1&=\text{Median of the first three value}=Y_{(\frac{3+1}{2})}\\&=Y_{(2)}=88.03 (\text{the second observation})\\
Q_3&=\text{Median of the last three values}=Y_{(\frac{3+5}{2})}\\
&=Y_{(4)}=94.90 (\text{the fourth observation})
\end{align*}

Quartiles in Statistics for Grouped Data

For the grouped data (in ascending order) the quartiles are calculated as:
\begin{align*}
Q_1&=l+\frac{h}{f}(\frac{n}{4}-c)\\
Q_2&=l+\frac{h}{f}(\frac{2n}{4}-c)\\
Q_3&=l+\frac{h}{f}(\frac{3n}{4}-c)
\end{align*}
where
$l$    is the lower class boundary of the class containing the $Q_1, Q_2$ or $Q_3$.
$h$    is the width of the class containing the $Q_1, Q_2$ or $Q_3$.
$f$    is the frequency of the class containing the $Q_1, Q_2$ or $Q_3$.
$c$    is the cumulative frequency of the class immediately preceding the class containing $Q_1, Q_2$ or $Q_3, \left[\frac{n}{4},\frac{2n}{4} \text{or} \frac{3n}{4}\right]$ are used to locate $Q_1, Q_2$ or $Q_3$ group.

Quartiles in Statistics: Relative Measure of Observation

Quartiles in Statistics Example: Find the quartiles for the following grouped data

Solution: To locate the class containing $Q_1$, find $\frac{n}{4}$th observation which is here $\frac{30}{4}$th observation i.e. 7.5th observation. Note that the 7.5th observation falls in the group ($Q_1$ group) 90.5–95.5.
\begin{align*}
Q_1&=l+\frac{h}{f}(\frac{n}{4}-c)\\
&=90.5+\frac{5}{4}(7.5-6)=90.3750
\end{align*}

For $Q_2$, the $\frac{2n}{4}$th observation=$\frac{2 \times 30}{4}$th observation = 15th observation falls in the group 95.5–100.5.
\begin{align*}
Q_2&=l+\frac{h}{f}(\frac{2n}{4}-c)\\
&=95.5+\frac{5}{10}(15-10)=98
\end{align*}

For $Q_3$, the $\frac{3n}{4}$th observation=$\frac{3\times 30}{4}$th = 22.5th observation. So
\begin{align*}
Q_3&=l+\frac{h}{f}(\frac{3n}{4}-c)\\
&=100.5+\frac{5}{6}(22.5-20)=102.5833
\end{align*}

Application of Quartiles

By analyzing quartiles, one can get insights into the:

  • Spread of the data: The distance between $Q_1$ and $Q_3$ (called the interquartile range or IQR) indicates how spread out the data is. A relatively large IQR indicates a wider distribution, while a small IQR shows that the data is more concentrated around the median ($Q_2$).
  • Presence of outliers: If the data points are extremely far from the quartiles, they might be outliers that could skew the analysis of measures like the mean.
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Range Measure of Dispersion (2013)

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Measure of Central Tendency provides typical value about the data set, but it does not tell the actual story about the data i.e. mean, median, and mode are enough to get summary information, though we know about the center of the data. In other words, we can measure the center of the data by looking at averages (mean, median, and mode). These measures tell nothing about the spread of data. So for more information about data, we need some other measure, such as the Range measure of dispersion or spread.

Range Measure of Dispersion

The Spread of data can be measured by calculating the range of data; the range tells us how many numbers of data extend. The range is an absolute measure of dispersion that can be found by subtracting the highest value (called upper bound) in data from the smallest value (called lower bound). i.e.

Range = Upper Bound – Lowest Bound
OR
Range = Largest Value – Smallest Value

This absolute measure of dispersion has disadvantages as range only describes the width of the data set (i.e. only spread out) measured in the same unit as data, but it does not give the real picture of how data is distributed. If data has outliers, using range to describe the spread of that can be very misleading as the range is sensitive to outliers.

We need to be careful in using the range measure of dispersion as it does not give the full picture of what’s going between the highest and lowest values. It might give a misleading picture of the spread of the data because it is based only on the two extreme values. Therefore, Range is an unsatisfactory measure of dispersion.

Range measure-of-dispersion

However, the range measure of dispersion is widely used in statistical process control such as control charts of manufactured products, daily temperature, stock prices, etc., applications as it is very easy to calculate. It is an absolute measure of dispersion, its relative measure known as the coefficient of dispersion defines the relation

\[Coefficient\,\, of\,\, Dispersion = \frac{x_m-x_0}{x_m-x_0}\]

Measure of Dispersion

The coefficient of dispersion is pure dimensionless and is used for comparison purposes.

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Absolute Measure of Dispersion

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An absolute Measure of Dispersion gives an idea about the amount of dispersion/ spread in a set of observations. These quantities measure the dispersion in the same units as the units of original data. The absolute measure of dispersion cannot be used to compare the variation of two or more series/ data sets. The absolute measure of dispersion does not in itself, tell whether the variation is large or small.

Absolute Measure of Dispersion

The absolute Measure of Dispersion:

  1. Range
  2. Quartile Deviation
  3. Mean Deviation
  4. Variance or Standard Deviation
Absolute Measures of Dispersion

Range

The Range is the difference between the largest value and the smallest value in the data set. For ungrouped data, let $X_0$ be the smallest value and $X_n$ be the largest  value in a data set then the range ($R$) is defined as
$R=X_n-X_0$.

For grouped data Range can be calculated in three different ways
R=Mid point of the highest class – Midpoint of the lowest class
R=Upper class limit of the highest class – Lower class limit of the lower class
R=Upper class boundary of the highest class – The lower class boundary of the lowest class

Quartile Deviation (Semi-Interquantile Range)

The Quartile deviation (an absolute measure of dispersion) is defined as the difference between the third and first quartiles, and half of this range is called the semi-interquartile range (SIQD) or simply quartile deviation (QD). $$QD=\frac{Q_3-Q_1}{2}$$

The Quartile Deviation is superior to the range as it is not affected by extremely large or small observations, anyhow it does not give any information about the position of observation lying outside the two quantities. It is not amenable to mathematical treatment and is greatly affected by sampling variability. Although Quartile Deviation is not widely used as a measure of dispersion, it is used in situations in which extreme observations are thought to be unrepresentative/ misleading. Quartile Deviation is not based on all observations therefore it is affected by extreme observations.

Note: The range “Median ± QD” contains approximately 50% of the data.

Mean Deviation (Average Deviation)

The Mean Deviation is another absolute measure of dispersion and is defined as the arithmetic mean of the deviations measured either from the mean or from the median. All these deviations are counted as positive to avoid the difficulty arising from the property that the sum of deviations of observations from their mean is zero.

$MD=\frac{\sum|X-\overline{X}|}{n}\quad$ for ungrouped data for mean
$MD=\frac{\sum f|X-\overline{X}|}{\sum f}\quad$ for grouped data for mean
$MD=\frac{\sum|X-\tilde{X}|}{n}\quad$ for ungrouped data for median
$MD=\frac{\sum f|X-\tilde{X}|}{\sum f}\quad$ for grouped data for median
Mean Deviation can be calculated about other central tendencies but it is least when deviations are taken as the median.

The Mean Deviation gives more information than the range or the Quartile Deviation as it is based on all the observed values. The Mean Deviation does not give undue weight to occasional large deviations, so it should likely be used in situations where such deviations are likely to occur.

Variance and Standard Deviation

This absolute measure of dispersion is defined as the mean of the squares of deviations of all the observations from their mean. Traditionally population variance is denoted by $\sigma^2$ (sigma square) and for sample data denoted by $S^2$ or $s^2$.

Symbolically
$\sigma^2=\frac{\sum(X_i-\mu)^2}{N}\quad$ Population Variance for ungrouped data
$S^2=\frac{\sum(X_i-\overline{X})^2}{n}\quad$ sample Variance for ungrouped data
$\sigma^2=\frac{\sum f(X_i-\mu)^2}{\sum f}\quad$ Population Variance for grouped data
$\sigma^2=\frac{\sum f (X_i-\overline{X})^2}{\sum f}\quad$ Sample Variance for grouped data

The variance is denoted by $Var(X)$ for random variable $X$. The term variance was introduced by R. A. Fisher (1890-1982) in 1918. The variance is in squares of units and the variance is a large number compared to observations themselves.
Note that there are alternative formulas to compute Variance or Standard Deviations.

The positive square root of the variance is called Standard Deviation (SD) to express the deviation in the same units as the original observation. It is a measure of the average spread about the mean and is symbolically defined as

$\sigma^2=\sqrt{\frac{\sum(X_i-\mu)^2}{N}}\quad$ Population Standard for ungrouped data
$S^2=\sqrt{\frac{\sum(X_i-\overline{X})^2}{n}}\quad$ Sample Standard Deviation for ungrouped data
$\sigma^2=\sqrt{\frac{\sum f(X_i-\mu)^2}{\sum f}}\quad$ Population Standard Deviation for grouped data
$\sigma^2=\sqrt{\frac{\sum f (X_i-\overline{X})^2}{\sum f}}\quad$ Sample Standard Deviation for grouped data
Standard Deviation is the most useful measure of dispersion and is credited with the name Standard Deviation by Karl Pearson (1857-1936).

In some text Sample, Standard Deviation is defined as $S^2=\frac{\sum (X_i-\overline{X})^2}{n-1}$ based on the argument that knowledge of any $n-1$ deviations determines the remaining deviations as the sum of n deviations must be zero. This is an unbiased estimator of the population variance $\sigma^2$. The Standard Deviation has a definite mathematical measure, it utilizes all the observed values and is amenable to mathematical treatment but affected by extreme values.

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Percentiles: Relative Standing

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Percentiles are a measure of the relative standing of observation within a data. Percentiles divide a set of observations into 100 equal parts, and percentile scores are frequently used to report results from national standardized tests such as NAT, GAT, and GRE, etc.

The $p$th percentile is the value $Y_{(p)}$ in order statistic such that $p$ percent of the values are less than the value $Y_{(p)}$ and $(100-p)$ (100-p) percent of the values are greater $Y_{(p)}$. The 5th percentile is denoted by $P_5$, the 10th by $P_{10}$ and 95th by $P_{95}$.

Percentiles for the Ungrouped data

To calculate percentiles (a measure of the relative standing of an observation) for the ungrouped data, adopt the following procedure:

  1. Order the observation
  2. For the $m$th percentile, determine the product $\frac{m.n}{100}$. If $\frac{m.n}{100}$ is not an integer, round it up and find the corresponding ordered value and if $\frac{m.n}{100}$ is an integer, say k, then calculate the mean of the $K$th and $(k+1)$th ordered observations.

Example: For the following height data collected from students find the 10th and 95th percentiles. 91, 89, 88, 87, 89, 91, 87, 92, 90, 98, 95, 97, 96, 100, 101, 96, 98, 99, 98, 100, 102, 99, 101, 105, 103, 107, 105, 106, 107, 112.

Solution: The ordered observations of the data are 87, 87, 88, 89, 89, 90, 91, 91, 92, 95, 96, 96, 97, 98, 98, 98, 99, 99, 100, 100, 101, 101, 102, 103, 105, 105, 106, 107, 107, 112.

\[P_{10}= \frac{10 \times 30}{100}=3\]

So the 10th percentile i.e. $P_{10}$ is the 3rd observation in sorted data is 88, which means that 10 percent of the observations in the data set are less than 88.

\[P_{95}=\frac{95 \times 30}{100}=28.5\]

The 29th observation is our 95th Percnetile i.e., $P_{95}=107$

Percentiles for the Frequency Distribution Table (Grouped data)

The $m$th percentile (a measure of the relative standing of an observation) for the Frequency Distribution Table (grouped data) is

\[P_m=l+\frac{h}{f}\left(\frac{m.n}{100}-c\right)\]

Like median, $\frac{m.n}{100}$ is used to locate the $m$th percentile group.

$l$    is the lower class boundary of the class containing the $m$th percentile
$h$   is the width of the class containing $P_m$
$f$    is the frequency of the class containing
$n$   is the total number of frequencies $P_m$
$c$    is the cumulative frequency of the class immediately preceding the class containing $P_m$

Note that the 50th percentile is the median by definition as half of the values in the data are smaller than the median and half of the values are larger than the median. Similarly, the 25th and 75th percentiles are the lower ($Q_1$) and upper quartiles ($Q_3$) respectively. The quartiles, deciles, and percentiles are also called quantiles or fractiles.

Percentiles: Measure of Relative Standing

Example: For the following grouped data compute $P_{10}$, $P_{25}$, $P_{50}$, and $P_{95}$ given below.Solution:

  1. Locate the 10th percentile (lower deciles i.e. $D_1$)by $\frac{10 \times n}{100}=\frac{10 \times 3o}{100}=3$ observation.
    so, $P_{10}$ group is 85.5–90.5 containing the 3rd observation
    \begin{align*}
    P_{10}&=l+\frac{h}{f}\left(\frac{10 n}{100}-c\right)\\
    &=85.5+\frac{5}{6}(3-0)\\
    &=85.5+2.5=88
    \end{align*}
  2. Locate the 25th percentile (lower quartiles i.e. $Q_1$)  by $\frac{10 \times n}{100}=\frac{25 \times 3o}{100}=7.5$ observation.
    so, $P_{25}$ group is 90.5–95.5 containing the 7.5th observation
    \begin{align*}
    P_{25}&=l+\frac{h}{f}\left(\frac{25 n}{100}-c\right)\\
    &=90.5+\frac{5}{4}(7.5-6)\\
    &=90.5+1.875=92.375
    \end{align*}
  3. Locate the 50th percentile (Median i.e. 2nd quartiles, 5th deciles) by $\frac{50 \times n}{100}=\frac{50 \times 3o}{100}=15$ observation.
    so, P50 group is 95.5–100.5 containing the 15th observation
    \begin{align*}
    P_{50}&=l+\frac{h}{f}\left(\frac{50 n}{100}-c\right)\\
    &=95.5+\frac{5}{10}(15-10)\\
    &=95.5+2.5=98
    \end{align*}
  4. Locate the 95th percentile by $\frac{95 \times n}{100}=\frac{95 \times 30}{100}=28.5$th observation.
    so, $P_{95}$ group is 105.5–110.5 containing the 3rd observation
    \begin{align*}
    P_{95}&=l+\frac{h}{f}\left(\frac{95 n}{100}-c\right)\\
    &=105.5+\frac{5}{3}(28.5-26)\\
    &=105.5+4.1667=109.6667
    \end{align*}

The percentiles and quartiles may be read directly from the graphs of the cumulative frequency function.

Further Reading: https://en.wikipedia.org/wiki/Percentile

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