Quartiles, percentiles and measures of dispersion Revision Notes

Quartiles split a data set into four equal parts, each containing 25% of the data. The first quartile (Q1) marks the 25th percentile, the second quartile (Q2) is the median or 50th percentile, and the third quartile (Q3) marks the 75th percentile. These values help describe the distribution and spread of data beyond just the median.

• Q1 = 25th percentile
• Q2 = 50th percentile (median)
• Q3 = 75th percentile

Start by arranging the data values in ascending order. The median (Q2) divides the data into two halves. To find Q1, calculate the median of the lower half (values below Q2). To find Q3, calculate the median of the upper half (values above Q2).

If the number of data points is odd, exclude the median when splitting the data into halves. If even, split the data evenly.

Example: Find quartiles for the data set: 7, 3, 9, 5, 12.

• Arrange: 3, 5, 7, 9, 12
• Median (Q2) = 7 (middle value)
• Lower half: 3, 5 → Q1 = median of 3 and 5 = (3+5)/2 = 4
• Upper half: 9, 12 → Q3 = median of 9 and 12 = (9+12)/2 = 10.5

An ungrouped frequency distribution lists distinct data values with their frequencies. To find quartiles, first calculate the cumulative frequency to determine the position of each quartile.

The position of Q1 is at 1/4 of the total frequency (n), Q2 at 1/2 of n, and Q3 at 3/4 of n.

Locate the data value corresponding to these positions by finding where the cumulative frequency reaches or exceeds these values.

• Calculate total frequency n = sum of frequencies
• Find positions: Q1 = n/4, Q2 = n/2, Q3 = 3n/4
• Use cumulative frequency to identify quartile values

Consider the data values and frequencies: 2 (3 times), 4 (5 times), 6 (4 times), 8 (2 times).

Total frequency n = 3 + 5 + 4 + 2 = 14. Positions: Q1 = 14/4 = 3.5, Q2 = 14/2 = 7, Q3 = 3×14/4 = 10.5.

Cumulative frequencies: 2 → 3, 4 → 8, 6 → 12, 8 → 14.

• Q1 position 3.5 lies between cumulative frequencies 3 and 8, so Q1 = 4
• Q2 position 7 lies between 3 and 8, so Q2 = 4
• Q3 position 10.5 lies between 8 and 12, so Q3 = 6

Quartiles provide insight into data spread and central tendency. The interquartile range (IQR), calculated as Q3 − Q1, measures the middle 50% spread, useful for identifying variability and outliers.

Understanding quartiles helps in comparing data sets and summarising distributions effectively.

• Quartiles divide data into four equal parts
• IQR = Q3 − Q1 measures spread of middle 50%
• Useful for detecting outliers and comparing distributions

Quartiles and percentiles are statistical tools used to describe the position of data points within a distribution.

Quartiles divide a data set into four equal parts, with the first quartile (Q1) marking the 25th percentile, the second quartile (Q2) the 50th percentile (also the median), and the third quartile (Q3) the 75th percentile.

Percentiles divide data into 100 equal parts, so the pth percentile is the value below which p% of the data lie. For example, the 90th percentile is the value below which 90% of the data fall.

• Quartiles: Q1 (25%), Q2 (50%, median), Q3 (75%)
• Percentiles: Divide data into 100 equal parts
• Used to understand data distribution and spread

A cumulative frequency graph plots cumulative frequency against the upper boundary of class intervals.

It shows the total number of observations less than or equal to a given value.

To construct one, first calculate cumulative frequencies from a grouped frequency distribution by adding frequencies successively from the lowest class upwards, then plot these against the upper class boundaries and join points smoothly or with straight lines (polygon).

• Cumulative frequency = sum of frequencies up to a class
• Graph plots cumulative frequency vs upper class boundary
• Useful for estimating quartiles and percentiles visually

Linear interpolation assumes data are evenly spread within a class interval or between points on a cumulative frequency graph.

To estimate a quartile or percentile, first find the cumulative frequency corresponding to that position (e.g., for Q1, find 25% of total frequency).

Locate this cumulative frequency on the vertical axis of the graph, then draw a horizontal line to meet the cumulative frequency curve or polygon, and from that point drop a vertical line to the horizontal axis to find the estimated value of the quartile or percentile.

• Calculate position: (percentile/100) × total frequency
• Use graph to find corresponding data value
• Assumes uniform distribution within intervals

When only grouped data are available, quartiles and percentiles can be estimated using linear interpolation within the relevant class interval.

Identify the class interval containing the quartile or percentile by locating the cumulative frequency just greater than the required position.

Apply the formula: Estimated value = lower class boundary + \[ (position - cumulative frequency before class) ÷ frequency in class \] × class width, where position is the cumulative frequency corresponding to the desired quartile or percentile.

• Find class containing desired quartile/percentile
• Use cumulative frequencies to locate position
• Apply linear interpolation formula within class

Linear interpolation can also estimate the number of observations within a particular range when data are grouped.

For example, to estimate how many observations lie between two percentiles, find the cumulative frequencies at those percentiles and subtract.

This method assumes uniform distribution within classes and is useful for understanding data spread and concentration in specific intervals.

• Calculate cumulative frequencies at given percentiles
• Subtract to find frequency in interval
• Assumes uniform distribution within classes

The range is a simple measure of dispersion that shows how spread out the values in a data set are. It is calculated by finding the difference between the largest and smallest values. This gives a quick sense of the total spread or variability in the data.

For example, if the data set is {4, 7, 9, 2, 5}, the smallest value is 2 and the largest is 9. The range is 9 − 2 = 7. This means the data values cover a span of 7 units.

• Range = Largest value − Smallest value
• Indicates the total spread of data values
• Easy to calculate but sensitive to extreme values

To calculate the range for a small data set, follow these steps:

1. List all the data values clearly, either in order or as given.

2. Identify the smallest (minimum) value in the set and the largest (maximum) value in the set carefully to avoid mistakes with negative numbers or decimals, if present.

• Example: Data set = {3, 8, 5, 10, 6}
• Smallest value = 3
• Largest value = 10
• Range = 10 − 3 = 7

An ungrouped frequency distribution lists distinct data values alongside their frequencies (how many times each value occurs). To find the range:

1. Identify the smallest data value with a non-zero frequency.

2. Identify the largest data value with a non-zero frequency.

• Example: Data values and frequencies:
• Value: 2, 4, 6, 8
• Frequency: 3, 5, 2, 1
• Smallest value = 2 (frequency 3)
• Largest value = 8 (frequency 1)
• Range = 8 − 2 = 6

Consider the data set: {12, 7, 15, 9, 10} Step 1: Identify smallest value = 7 Step 2: Identify largest value = 15 Step 3: Calculate range = 15 − 7 = 8 The range is 8, meaning the data values span 8 units.

• Check carefully for the correct minimum and maximum values
• Subtract minimum from maximum to find range

Given the frequency distribution: Value: 5, 7, 9, 11 Frequency: 4, 2, 3, 1 Step 1: Smallest value with frequency > 0 is 5 Step 2: Largest value with frequency > 0 is 11 Step 3: Calculate range = 11 − 5 = 6

• Always confirm frequencies are positive to avoid including values not in the data
• Range shows total spread of data values

While the range is easy to calculate and understand, it has some limitations:

It only considers the two extreme values and ignores the distribution of the rest of the data.

It is sensitive to outliers; a single very high or low value can greatly increase the range, giving a misleading impression of spread.

• Use range alongside other measures of dispersion for fuller understanding
• Be cautious when data contains outliers or extreme values

Quartiles divide an ordered data set into four equal parts. The lower quartile (Q1) marks the 25th percentile, the median (Q2) the 50th percentile, and the upper quartile (Q3) the 75th percentile. The interquartile range (IQR) measures the spread of the middle 50% of data by subtracting Q1 from Q3: IQR = Q3 - Q1.

Unlike the range, which uses the extreme values, the IQR is less sensitive to outliers and gives a better sense of typical variability in the data.

• Quartiles split data into four equal parts.
• Q1 is the median of the lower half, Q3 is the median of the upper half.
• IQR = Q3 - Q1 measures spread of the central 50% of data.
• IQR is robust against extreme values.

Start by arranging the data in ascending order. Find the median (Q2) to split the data into two halves. If the number of data points is odd, exclude the median when splitting; if even, split evenly.

The lower quartile Q1 is the median of the lower half, and the upper quartile Q3 is the median of the upper half.

Example: For data set 3, 7, 8, 12, 13, 14, 18, 21, 23: Sorted order is the same. Median (Q2) is 13 (middle value). Lower half: 3, 7, 8, 12; median of lower half (Q1) is (7+8)/2 = 7.5. Upper half: 14, 18, 21, 23; median of upper half (Q3) is (18+21)/2 = 19.5. Then IQR = 19.5 - 7.5 = 12.

• Arrange data in ascending order.
• Find median to split data into halves.
• Calculate Q1 as median of lower half.
• Calculate Q3 as median of upper half.
• Subtract Q1 from Q3 to get IQR.

An ungrouped frequency distribution lists distinct data values with their frequencies. To find quartiles, first expand the data by repeating each value according to its frequency, then arrange in ascending order.

Alternatively, use cumulative frequencies to locate quartile positions without full expansion.

Example: Data values 2, 4, 5 with frequencies 3, 2, 5 respectively. Expanded data: 2, 2, 2, 4, 4, 5, 5, 5, 5, 5 (10 values). Median position: (10+1)/2 = 5.5th value between 4 and 4, so median = 4. Lower half: first 5 values (2,2,2,4,4), Q1 is median of these: 3rd value = 2. Upper half: last 5 values (4,5,5,5,5), Q3 is median: 3rd value = 5. IQR = 5 - 2 = 3.

• Use frequencies to expand data or use cumulative frequencies.
• Find total number of data points (N).
• Locate quartile positions: Q1 at 0.25(N+1), Q3 at 0.75(N+1).
• Identify Q1 and Q3 values from ordered data.
• Calculate IQR = Q3 - Q1.

The IQR quantifies the spread of the middle 50% of data, showing how concentrated or dispersed the central values are.

A small IQR indicates data are closely clustered around the median, while a large IQR shows greater variability.

Because it ignores the lowest 25% and highest 25% of data, the IQR is not influenced by extreme values or outliers, making it a reliable measure of spread for skewed distributions.

• IQR measures spread of central 50% of data.
• Smaller IQR means less variability.
• IQR is robust to outliers and skewed data.
• Useful for comparing variability between data sets.

1. Arrange data in ascending order (expand frequencies if needed).

2. Find the median to split data into lower and upper halves.

3. Find Q1 as median of lower half and Q3 as median of upper half (exclude median if odd number of data points). 4. Calculate IQR = Q3 - Q1.

• Order data first.
• Split data at median.
• Find medians of halves for Q1 and Q3.
• Subtract Q1 from Q3 for IQR.

The interquartile range (IQR) is a measure of statistical dispersion that describes the spread of the middle 50% of a data set. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1). Q1 is the value below which 25% of the data lie, and Q3 is the value below which 75% of the data lie. The IQR is useful because it is not affected by extreme values and gives a clear picture of the central concentration of data points.

• IQR = Q3 − Q1
• Represents the range of the middle 50% of data
• Less sensitive to outliers than the full range

A cumulative frequency graph plots cumulative frequencies against the upper boundaries of class intervals. To estimate Q1 and Q3 from such a graph, first calculate the total number of observations (n). Then find the cumulative frequencies corresponding to 25% and 75% of n, which are 0.25n and 0.75n respectively. On the graph, locate these cumulative frequencies on the vertical axis, draw horizontal lines to meet the curve, and then drop vertical lines to the horizontal axis to read off the corresponding data values. These values are the estimates of Q1 and Q3.

• Calculate 0.25n and 0.75n for total observations n
• Use the cumulative frequency graph to find corresponding data values
• These values estimate Q1 and Q3

Often, the cumulative frequency values 0.25n or 0.75n do not fall exactly on a plotted point on the graph or at a class boundary in grouped data. Linear interpolation assumes that data are evenly distributed within the interval and estimates the quartile value proportionally between two known points. The formula for linear interpolation between two points (x1, y1) and (x2, y2) to find x corresponding to y is:

\[ x = x_1 + \frac{(y - y_1)}{(y_2 - y_1)} \times (x_2 - x_1) \]

This method is applied both when reading from cumulative frequency graphs and when estimating quartiles from grouped frequency tables.

• Assumes uniform distribution within intervals
• Use formula to estimate quartile values between known points
• Improves accuracy of quartile estimation

In grouped frequency distributions, data are presented in class intervals with frequencies. To estimate Q1 or Q3, first calculate the cumulative frequencies for each class. Identify the class interval containing the quartile by finding where the cumulative frequency reaches or exceeds 0.25n for Q1 or 0.75n for Q3. Then apply linear interpolation within that class interval to estimate the quartile value. This involves using the lower class boundary, the cumulative frequency before the class, the frequency of the class, and the class width.

• Calculate cumulative frequencies for classes
• Locate class containing Q1 or Q3 based on 0.25n or 0.75n
• Apply linear interpolation within that class interval

Once Q1 and Q3 have been estimated, the interquartile range is found by subtracting Q1 from Q3. This value quantifies the spread of the middle half of the data. For example, if Q1 = 12 and Q3 = 20, then IQR = 20 − 12 = 8. The IQR is expressed in the same units as the original data and provides insight into data variability without being influenced by extreme values.

• IQR = Q3 − Q1
• Expressed in original data units
• Indicates spread of central 50% of data

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of numerical data. A low standard deviation means the data points tend to be close to the mean (average), while a high standard deviation indicates the data are spread out over a wider range of values.

It is important to distinguish standard deviation from other measures of spread such as range or interquartile range because it takes into account all data points and their distances from the mean, providing a more comprehensive measure of variability.

• Standard deviation measures how spread out data values are around the mean.
• It uses all data points, not just extremes.
• It is expressed in the same units as the data.

For a small data set, the standard deviation \( \sigma \) is calculated using the formula:

\[ \sigma = \sqrt{ \frac{\sum (x - \bar{x})^2}{n} } \]

where: \( x \) represents each data value, \( \bar{x} \) is the mean of the data, and \( n \) is the number of data points. The steps are: calculate the mean, find each deviation from the mean, square these deviations, sum them, divide by \( n \), then take the square root.

• Calculate the mean \( \bar{x} \).
• Find each deviation \( x - \bar{x} \).
• Square each deviation.
• Sum all squared deviations.
• Divide by \( n \) (number of data points).
• Take the square root to find \( \sigma \).

Consider the data set: 4, 7, 8, 6, 5. Calculate the standard deviation.

Step 1: Calculate the mean: \( \bar{x} = \frac{4+7+8+6+5}{5} = \frac{30}{5} = 6 \).

Step 2: Calculate each deviation and square: (4-6)²=4, (7-6)²=1, (8-6)²=4, (6-6)²=0, (5-6)²=1. Sum = 4+1+4+0+1=10. Step 3: Divide by \( n=5 \): \( \frac{10}{5} = 2 \). Step 4: Square root: \( \sqrt{2} \approx 1.414 \). The standard deviation is approximately 1.414 units.

• Mean is 6 units.
• Sum of squared deviations is 10.
• Variance is 2.
• Standard deviation is \( \sqrt{2} \approx 1.414 \).

When data are presented as an ungrouped frequency distribution, each distinct value \( x_i \) has a frequency \( f_i \). The formula adapts to:

\[ \sigma = \sqrt{ \frac{\sum f_i (x_i - \bar{x})^2}{N} } \]

where \( N = \sum f_i \) is the total number of data points, and \( \bar{x} = \frac{\sum f_i x_i}{N} \) is the mean. The process is similar but accounts for frequencies by weighting each squared deviation.

• Calculate the mean using frequencies.
• Calculate squared deviations weighted by frequencies.
• Sum weighted squared deviations.
• Divide by total frequency \( N \).
• Take the square root for standard deviation.

Data values and frequencies: \( x: 2, 4, 6 \); frequencies \( f: 3, 2, 5 \).

Step 1: Calculate total frequency: \( N = 3+2+5=10 \).

Step 2: Calculate mean: \( \bar{x} = \frac{3\times2 + 2\times4 + 5\times6}{10} = \frac{6 + 8 + 30}{10} = \frac{44}{10} = 4.4 \). Step 3: Calculate weighted squared deviations: \( 3(2-4.4)^2 = 3(5.76) = 17.28 \), \( 2(4-4.4)^2 = 2(0.16) = 0.32 \), \( 5(6-4.4)^2 = 5(2.56) = 12.8 \). Sum = 17.28 + 0.32 + 12.8 = 30.4. Step 4: Divide by \( N=10 \): 3.04. Step 5: Square root: \( \sqrt{3.04} \approx 1.744 \). The standard deviation is approximately 1.744 units.

• Total frequency is 10.
• Mean is 4.4 units.
• Sum of weighted squared deviations is 30.4.
• Variance is 3.04.
• Standard deviation is approximately 1.744.

For grouped data, exact values are not known, only class intervals and frequencies. To estimate standard deviation:

1. Find the midpoint \( x_i \) of each class interval.

2. Use these midpoints as representative values for the class data points in calculations, treating the data as an ungrouped frequency distribution with values \( x_i \) and frequencies \( f_i \). The formula is: \[ \sigma \approx \sqrt{ \frac{\sum f_i (x_i - \bar{x})^2}{N} } \] where \( \bar{x} = \frac{\sum f_i x_i}{N} \). This is an approximation because actual data within classes may vary.

• Calculate midpoints of each class.
• Calculate mean using midpoints and frequencies.
• Calculate weighted squared deviations using midpoints.
• Divide by total frequency.
• Take square root for estimated standard deviation.

Measures of dispersion quantify how spread out data values are in a dataset. While measures of central tendency (like mean or median) describe the centre, dispersion measures reveal variability, which is crucial for understanding data reliability and diversity. The four common measures are range, interquartile range (IQR), variance, and standard deviation. Each has distinct properties affecting their usefulness depending on the data and purpose.

• Dispersion shows variability or spread in data.
• Key measures: range, IQR, variance, standard deviation.
• Choice depends on data characteristics and analysis needs.

The range is the difference between the maximum and minimum values in a dataset. It is the simplest measure of dispersion and easy to calculate, requiring only the two extreme values. For example, if test scores range from 40 to 85, the range is 85 - 40 = 45.

However, the range is highly sensitive to outliers. A single unusually high or low value can greatly increase the range, giving a misleading impression of variability. It also ignores all other data points, so it does not reflect the overall spread well when data are skewed or contain anomalies.

• Range = maximum value − minimum value.
• Example: Data {40, 55, 60, 70, 85} → Range = 85 − 40 = 45.
• Sensitive to outliers and ignores data distribution.

The interquartile range measures the spread of the middle 50% of data, calculated as the difference between the third quartile (Q3) and the first quartile (Q1). For example, if Q3 = 75 and Q1 = 50, then IQR = 75 − 50 = 25.

Because it focuses on the central portion of data, the IQR is less affected by extreme values or outliers. This makes it a better choice when data contain anomalies or are skewed. It also complements the median as a measure of central tendency.

However, the IQR ignores the spread of the lowest 25% and highest 25% of data, so it does not capture total variability.

• IQR = Q3 − Q1, covering the middle 50% of data.
• Example: Q1 = 50, Q3 = 75 → IQR = 25.
• Robust against outliers, useful for skewed data.
• Does not reflect full data spread.

Variance measures the average of the squared differences between each data point and the mean. It is calculated by:

\[ \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{n} \]

where \(x_i\) are data values, \(\bar{x}\) is the mean, and \(n\) is the number of data points. For example, for data {2, 4, 6}, mean is 4, squared differences are (2−4)²=4, (4−4)²=0, (6−4)²=4, sum is 8, variance = 8/3 ≈ 2.67 units². The squared units can be difficult to interpret directly because they are not in the original data units (e.g., squared cm instead of cm). Variance is sensitive to outliers because squaring amplifies large deviations.

• Variance = average squared deviation from mean.
• Example: Data {2,4,6}, variance ≈ 2.67 units².
• Units are squared, less intuitive.
• Sensitive to outliers due to squaring.

Standard deviation is the square root of variance, returning the measure to the original data units. Using the previous example, standard deviation = \(\sqrt{2.67} \approx 1.63\) units.

It is widely used because it provides a clear sense of typical deviation from the mean and is mathematically convenient for further analysis, such as in normal distribution assumptions.

Like variance, it is sensitive to outliers and best used when data are roughly symmetric without extreme values.

• Standard deviation = \(\sqrt{\text{variance}}\).
• Example: SD ≈ 1.63 units for data {2,4,6}.
• Expressed in original units, easier to interpret.
• Sensitive to outliers, assumes symmetric data.

The choice depends on data characteristics and the analysis goal:

If data have outliers or are skewed, the IQR is preferred because it is robust and focuses on the central data.

For quick, simple spread estimation, the range can be used but with caution if outliers exist, as it may exaggerate variability or be misleading in small samples or skewed data sets. When data are approximately symmetric and outliers are minimal, standard deviation is preferred because it relates directly to the mean and is useful in many statistical methods. Variance is mainly used in theoretical contexts or as an intermediate step to calculate standard deviation, but its squared units limit direct interpretation. Justifying the choice involves explaining these considerations clearly in context, for example: “Because the dataset contains extreme values, the IQR is chosen to summarise spread as it is not distorted by outliers.”

• Use IQR for skewed data or outliers.
• Use range for simple, quick spread but beware outliers.
• Use standard deviation for symmetric data without outliers.
• Justify choice based on data features and analysis purpose.