For the 21 readings 2.48, 2.44, 2.40, 2.42, 2.38, 2.49, 2.39, 2.51, 2.40, 2.40, 2.49, 2.43, 2.47, 2.47, 2.49, 2.48, 2.45, 2.35, 2.45, 2.43, 2.43, find the mean, median, mode, range, variance and standard deviation.

Step-by-step solution

1. 1

   Total the values and divide by n. There are $n = 21$ readings and they add to

   $\sum x = 51.25$

   so

   $\bar{x} = \frac{51.25}{21} = \frac{205}{84} \approx 2.4405$

   Add the column twice — a single misread digit here (for instance totalling $51.45$) shifts the mean and every squared deviation that follows.

2. 2

   Sort the data once; median, mode and range all read off the sorted list.

   $2.35,\ 2.38,\ 2.39,\ 2.40,\ 2.40,\ 2.40,\ 2.42,\ 2.43,\ 2.43,\ 2.43,\ \mathbf{2.44},$
   $2.45,\ 2.45,\ 2.47,\ 2.47,\ 2.48,\ 2.48,\ 2.49,\ 2.49,\ 2.49,\ 2.51$

3. 3

   Median and range. With $21$ values (odd) the median is the single middle entry, position $\tfrac{21+1}{2} = 11$:

   $\text{median} = 2.44$

   and

   $\text{range} = 2.51 - 2.35 = 0.16$

4. 4

   Mode: check for ties before answering. Counting frequencies, $2.40$, $2.43$ and $2.49$ each occur three times and nothing occurs more often, so the set is trimodal:

   $\text{modes} = 2.40,\ 2.43,\ 2.49$

   A data set can legitimately have several modes; picking just one would be wrong here.

5. 5

   Sum the squared deviations from the exact mean. Using $\bar{x} = 205/84$ rather than a rounded $2.44$:

   $\sum (x - \bar{x})^2 = \frac{979}{26250} \approx 0.037295$

6. 6

   Divide and take the square root. Treating the readings as a sample ($n - 1 = 20$):

   $s^2 = \frac{0.037295}{20} \approx 0.0018648, \qquad s = \sqrt{0.0018648} \approx 0.0432$

   As a population ($n = 21$) instead: $\sigma^2 \approx 0.0017760$ and $\sigma \approx 0.0421$. Either way the spread is about $0.04$, roughly a quarter of the range, which is typical for a tightly clustered set.