krit.club logo

Statistics and Probability - Measures of central tendency – Measures of spread

Grade 11IB_AA

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Measures of central tendency identify the 'center' of a data distribution. The three main measures are the mean (xˉ\bar{x}), the median, and the mode.

•

The arithmetic mean is the sum of all values divided by the total number of values (nn). It is sensitive to extreme outliers.

•

The median is the middle value when data is arranged in ascending order. If nn is even, it is the average of the two middle values.

•

The mode is the value that occurs most frequently in a data set. A set can be bimodal if two values share the highest frequency.

•

Measures of spread (dispersion) describe how far the data points are distributed from the center.

•

The Range is the difference between the maximum and minimum values: Range=xmax−xminRange = x_{max} - x_{min}.

•

Quartiles divide the data into four equal parts. Q1Q_1 is the lower quartile (25th percentile), and Q3Q_3 is the upper quartile (75th percentile).

•

The Interquartile Range (IQRIQR) is the difference between the upper and lower quartiles (Q3−Q1Q_3 - Q_1), representing the spread of the middle 50% of the data.

•

Standard deviation (σ\sigma) measures the average distance of each data point from the mean. A low σ\sigma indicates data is clustered closely around the mean.

•

Variance (σ2\sigma^2) is the square of the standard deviation.

•

An outlier is typically defined as any value xx such that x>Q3+1.5×IQRx > Q_3 + 1.5 \times IQR or x<Q1−1.5×IQRx < Q_1 - 1.5 \times IQR.

📐Formulae

xˉ=∑i=1nxin\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}

xˉ=∑fixi∑fi\bar{x} = \frac{\sum f_i x_i}{\sum f_i}

IQR=Q3−Q1IQR = Q_3 - Q_1

σ=∑i=1kfi(xi−xˉ)2n\sigma = \sqrt{\frac{\sum_{i=1}^{k} f_i (x_i - \bar{x})^2}{n}}

σ2=∑i=1kfi(xi−xˉ)2n\sigma^2 = \frac{\sum_{i=1}^{k} f_i (x_i - \bar{x})^2}{n}

💡Examples

Problem 1:

Given the data set: 4,8,2,9,12,5,84, 8, 2, 9, 12, 5, 8. Calculate the mean, median, and range.

Solution:

  1. Arrange the data: 2,4,5,8,8,9,122, 4, 5, 8, 8, 9, 12.
  2. Mean: xˉ=2+4+5+8+8+9+127=487≈6.86\bar{x} = \frac{2+4+5+8+8+9+12}{7} = \frac{48}{7} \approx 6.86.
  3. Median: The 4th value is 88.
  4. Range: 12−2=1012 - 2 = 10.

Explanation:

To find the median, the data must be sorted. The mean is the average of all values, and the range is the total spread.

Problem 2:

For a data set where Q1=15Q_1 = 15 and Q3=25Q_3 = 25, determine the IQRIQR and identify if the value 4242 is an outlier.

Solution:

  1. IQR=Q3−Q1=25−15=10IQR = Q_3 - Q_1 = 25 - 15 = 10.
  2. Outlier boundary: Q3+1.5×IQR=25+1.5(10)=25+15=40Q_3 + 1.5 \times IQR = 25 + 1.5(10) = 25 + 15 = 40.
  3. Since 42>4042 > 40, the value 4242 is an outlier.

Explanation:

The Interquartile Range measures the spread of the middle data. The 1.5×IQR1.5 \times IQR rule is the standard method for detecting outliers in IB mathematics.

Problem 3:

Calculate the standard deviation for the following values: 2,4,62, 4, 6.

Solution:

  1. xˉ=2+4+63=4\bar{x} = \frac{2+4+6}{3} = 4.
  2. Calculate (xi−xˉ)2(x_i - \bar{x})^2: (2−4)2=4(2-4)^2 = 4 (4−4)2=0(4-4)^2 = 0 (6−4)2=4(6-4)^2 = 4
  3. Variance σ2=4+0+43=83≈2.67\sigma^2 = \frac{4+0+4}{3} = \frac{8}{3} \approx 2.67.
  4. σ=83≈1.63\sigma = \sqrt{\frac{8}{3}} \approx 1.63.

Explanation:

The standard deviation is found by calculating the mean, finding the squared deviations from that mean, averaging them to get variance, and then taking the square root.