Measures of central tendency , review

A measure of central tendency is a single number that summarises a data set's "centre". We use three throughout the chapter: mean, median, mode. You met these in earlier classes; here we collect the formulas in the notation we will need for the rest of the chapter.

The three measures

Let the data be $x_1, x_2, \dots, x_n$ (raw) or organised in a frequency table $(x_i, f_i)$.

Arithmetic mean

For raw data: $\bar{x} = \dfrac{\sum x_i}{n}$.

For frequency data: $\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i} = \dfrac{\sum f_i x_i}{N}$, where $N = \sum f_i$.

For grouped (class-interval) data, replace each class by its midpoint $x_i$ and use the frequency formula.

Median

Arrange the data in non-decreasing order. The median is the middle value:

• If $n$ is odd: $M = x_{(n+1)/2}$.
• If $n$ is even: $M = \dfrac{x_{n/2} + x_{n/2 + 1}}{2}$.

For grouped data, locate the median class (the class containing the $N/2$-th observation) and use the interpolation formula: $M = \ell + \frac{N/2 - F}{f} \cdot h,$ where $\ell$ is the lower boundary of the median class, $F$ is the cumulative frequency before that class, $f$ is its frequency, and $h$ is the class width.

Mode

The mode is the value (or class) of highest frequency. For grouped data: $\text{Mode} = \ell + \frac{f_1 - f_0}{2 f_1 - f_0 - f_2} \cdot h,$ where $\ell$ is the lower boundary of the modal class, $f_1$ is the modal frequency, $f_0$ and $f_2$ are the frequencies of the preceding and following classes.

Relationship (empirical)

For moderately skewed distributions: $\text{Mode} \approx 3 \cdot \text{Median} - 2 \cdot \text{Mean}.$

Properties of the mean

Shift. If each $x_i$ is replaced by $x_i + a$, then $\bar{x}$ shifts by $a$.

Scale. If each $x_i$ is replaced by $b \cdot x_i$, then $\bar{x}$ multiplies by $b$.

Sum of deviations. $\sum (x_i - \bar{x}) = 0$ , by definition of the mean.

Worked examples

Example 1. Find the mean of $5, 7, 9, 11, 13$.

$\bar{x} = (5 + 7 + 9 + 11 + 13)/5 = 45/5 = 9$.

Example 2. Find the median of $3, 7, 8, 11, 14, 16, 18, 20$.

$n = 8$ (even). Median = $(x_4 + x_5)/2 = (11 + 14)/2 = 12.5$.

Example 3. Find the mean of the frequency data:

$N = 15$. $\sum f x = 2 + 6 + 15 + 16 + 5 = 44$. $\bar{x} = 44/15 \approx 2.93$.

Example 4. Find the mode of $4, 5, 6, 6, 7, 6, 8, 5, 9, 6$.

Count: $4(1), 5(2), 6(4), 7(1), 8(1), 9(1)$. Mode is $6$.

Example 5. Estimate the median of the grouped data:

$N = 30$, $N/2 = 15$. Cumulative: $5, 17, 27, 30$. The $15$-th observation is in class $10$–$20$. Apply formula: $M = 10 + (15 - 5)/12 \cdot 10 = 10 + 100/12 \approx 18.33$.

Try it yourself

1. Mean of $10, 20, 30, 40, 50$.
2. Median of $2, 5, 7, 9, 11, 14$.
3. Mode of $1, 2, 2, 3, 3, 3, 4, 4, 5$.
4. The mean of $5$ numbers is $20$. If a sixth number is added, mean becomes $25$. Find the new number.
5. The mean of $10$ items is $50$. One item recorded as $42$ should be $52$. Find the correct mean.
6. For the frequency table $x: 10, 20, 30, 40, 50$ with frequencies $3, 5, 8, 6, 3$, find the mean.
7. Find the median of $1, 2, 3, 4, 5, 6, 7, 8, 9$.
8. If $\bar{x} = 25$ and a new observation $40$ is added, the new mean of $n + 1$ numbers becomes $26$. Find $n$.
9. Estimate mode of the grouped data: classes $0$–$5, 5$–$10, 10$–$15, 15$–$20$ with frequencies $4, 9, 6, 2$.
10. Find the mean of the first $10$ odd natural numbers.
11. The marks of $5$ students are $40, 50, 60, x, 80$. The mean is $60$. Find $x$.
12. For data values $a, a+d, a+2d, \dots, a+(n-1)d$ (an AP), show that the mean is $a + (n-1)d/2$.

Pitfalls / Tricks

• Mean is sensitive to outliers; median is not.
• For grouped data, use the midpoint of each class as $x_i$.
• $\sum (x_i - \bar{x}) = 0$ always , this is a useful check.
• Insight. Central tendency tells you "where" the data is. Dispersion (next subtopic) tells you "how spread out" , equally important.