IB Mathematics AI SL Measures of central tendency MAI Study Notes - New Syllabus
IB Mathematics AI SL Measures of central tendency MAI Study Notes
LEARNING OBJECTIVE
- Measures of central tendency (mean, median and mode).
Key Concepts:
- Outliers
- Univariate Data
- Interpreting Data
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MEASURES OF CENTRAL TENDENCY (3M)
Measures of Central Tendency (3M)
Definition:
These are values that describe the center or typical value of a data set.
Mean (Arithmetic Mean):
Sum of all data values divided by the number of values.
Formula:
\( \bar{x} = \frac{\sum x_i}{n} \) where \(x_i\) = each data point, \(n\) = number of data points.
Median:
The middle value when data is arranged in ascending order.
If \(n\) is odd, median is \(\frac{n+1}{2}\)-th value; if even, average of \(\frac{n}{2}\)-th and \(\frac{n}{2} + 1\)-th values.
Mode:
The most frequently occurring value(s) in the data set.
Example Calculate the mean of the data set: 4, 8, 6, 5, 7. ▶️ Answer/ExplanationSolution: Sum of data $= 4 + 8 + 6 + 5 + 7 = 30$ Number of data points $= 5$ Mean = Total sum $÷$ Number of data $= 30 ÷ 5 = 6$ |
Example Find the median of the data set: 12, 7, 9, 15, 10. ▶️ Answer/ExplanationSolution: Order data from smallest to largest: 7, 9, 10, 12, 15 Since there are 5 data points (odd number), median is the middle value: 10 |
Example Find the mode of the data set: 5, 7, 5, 8, 9, 5, 7. ▶️ Answer/ExplanationSolution: Count the frequency of each number:
Mode = 5 (most frequent) |
ESTIMATION OF MEAN FROM GROUPED DATA
Estimation of Mean from Grouped Data
Definition:
When data is grouped into class intervals, mean is estimated using midpoints.
Formula:
\( \displaystyle \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \)
Where:
\(f_i\) = frequency of the \(i\)-th class
\(x_i\) = midpoint of the \(i\)-th class
Steps:
Find midpoints for each class interval.
Multiply each midpoint by its class frequency.
Sum all products and divide by total frequency.
Example: Measures of Central Tendency and Spread from Grouped Frequency Table Consider the data: 10, 20, 20, 20, 30, 30, 40, 50, 70, 70, 80 (total \( n = 11 \)). The frequency table is:
Calculate the mean, mode, median, standard deviation, range, and interquartile range. ▶️ Answer/ExplanationSolution: Mean: $ \text{mean} = \frac{1 \times 10 + 3 \times 20 + 2 \times 30 + 1 \times 40 + 1 \times 50 + 2 \times 70 + 1 \times 80}{11} = \frac{440}{11} = 40 $ Mode: The value with highest frequency is 20 (frequency = 3). Median: Position \( \frac{n+1}{2} = 6 \). Using cumulative frequencies (see below): the 6th entry corresponds to 30, so median = 30.
Standard Deviation: From a calculator or GDC, \( \sigma \approx 22.96 \). Range: $ \text{range} = \text{max} – \text{min} = 80 – 10 = 70 $ Interquartile Range (IQR): \( Q_1 \) = median of first 5 entries (position 3) = 20. |
MODAL CLASS (FOR EQUAL CLASS INTERVALS ONLY)
Modal Class (For Equal Class Intervals Only)
Definition:
The modal class is the class interval with the highest frequency in grouped data with equal class widths.
Interpretation: ‘
This class represents the most common range of data values.
Example: Suppose 100 students took an exam with scores between 1 and 60. The grouped frequency table is:
Find the mean, standard deviation, and modal class. ▶️ Answer/ExplanationSolution: Step 1: Use midpoints of each interval. $ \mu = \frac{8 \times 5 + 12 \times 15 + 10 \times 25 + 25 \times 35 + 35 \times 45 + 10 \times 55}{100} = \frac{3470}{100} = 34.7 $ Step 2: Standard Deviation (from GDC): $ \sigma = 14.31 $ Step 3: Modal Class: The class interval with the highest frequency is: $ \text{Modal class} = 40 < x \leq 50 $ |
Example: (USING GDC) Consider the frequency histogram for the distribution of the duration, t, in minutes of meeting times
Find an estimate for the mean time. ▶️ Answer/ExplanationSolution: Each class will be represented by its midpoint. So an estimate of the mean is $\bar{x} = \frac{\sum x_i}{n} \approx 18.077$ where $x_i$ is the midpoint of the ith class.
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MEASURES OF DISPERSION
Measures of Dispersion
Dispersion measures how spread out data values are around the central tendency.
Interquartile Range (IQR): Difference between third quartile (Q3) and first quartile (Q1).
\( \mathrm{IQR} = Q_3 – Q_1 \)
Standard Deviation (SD): Square root of variance, represents average distance of data points from the mean.
For sample data:
\( s = \sqrt{\frac{1}{n-1} \sum (x_i – \bar{x})^2} \)
Variance: Average of squared deviations from the mean.
\( s^2 = \frac{1}{n-1} \sum (x_i – \bar{x})^2 \)
USE OF GDC
We can use the GDC to obtain all these measures. For Casio CFX:
MENU → STAT → Enter data in List 1 → CALC → 1VAR → Obtain statistics.
Example Calculate the range of the data set: 14, 22, 19, 17, 24. ▶️ Answer/ExplanationSolution: Maximum value = 24 Minimum value = 14 Range = Maximum – Minimum = 24 – 14 = 10 |
Example Find the variance of the data set: 2, 4, 6, 8, 10. ▶️ Answer/ExplanationSolution: Calculate the mean: (2 + 4 + 6 + 8 + 10) ÷ 5 = 30 ÷ 5 = 6 Calculate squared differences from mean:
Find the average of squared differences (variance): (16 + 4 + 0 + 4 + 16) ÷ 5 = 40 ÷ 5 = 8 |
EFFECT OF CONSTANT CHANGES ON DATA
Effect of Constant Changes on Data
When data is transformed by adding, subtracting, multiplying, or dividing by constants, the measures of central tendency and spread change in predictable ways:
- Adding or subtracting a constant:
- The mean increases or decreases by the same constant.
- The standard deviation and variance remain unchanged because spread between values does not change.
- Multiplying or dividing by a constant:
- The mean is multiplied or divided by that constant.
- The standard deviation and variance are also multiplied or divided by the absolute value of the constant.
Examples:
- If 3 is subtracted from every data point, the mean decreases by 3, but the standard deviation stays the same.
- If all data points are doubled, both the mean and standard deviation are doubled.
Example Suppose the data set is: 5, 8, 10, 12, 15. Find the mean and standard deviation. Then find the new mean and standard deviation if 4 is subtracted from each data point. ▶️ Answer/ExplanationSolution: Original mean: \( \frac{5+8+10+12+15}{5} = \frac{50}{5} = 10 \) Original standard deviation: Calculate variance first, then sqrt (approx.) 3.69 New data set after subtracting 4: 1, 4, 6, 8, 11 New mean: \( 10 – 4 = 6 \) New standard deviation: unchanged, approx. 3.69 Thus, subtracting a constant shifts the mean but does not affect the spread (standard deviation). |
QUARTILES OF DISCRETE DATA
Quartiles of Discrete Data
Definition:
Quartiles divide ordered data into four equal parts.
- Q1 (First Quartile): Median of the lower half of the data (25th percentile).
- Q2 (Second Quartile): Median of the entire data set (50th percentile).
- Q3 (Third Quartile): Median of the upper half of the data (75th percentile).
Use:
Helps describe the spread and skewness of data.
Example: Suppose 100 students took an exam with scores between 1 and 60. The grouped frequency table is:
Determine the quartiles, draw a box plot, and identify any outliers. ▶️Answer/ExplanationSolution: Use cumulative frequency diagram.
Draw Box and Whisker Plot
Calculate IQR and check for outliers $ \text{IQR} = Q_3 – Q_1 = 46 – 25 = 21 $ $ \text{Lower boundary} = Q_1 – 1.5 \times \text{IQR} = 25 – 1.5 \times 21 = -6.5 $ $ \text{Upper boundary} = Q_3 + 1.5 \times \text{IQR} = 46 + 1.5 \times 21 = 77.5 $ Conclusion: No outliers exist in the data set. |
