1. **State the problem:** We are given a data set and asked to check the calculations of mean, median, range, variance, and standard deviation. 2. **Mean calculation:** The mean $\bar{X}$ is the sum of all data points divided by the number of points. $$\bar{X} = \frac{175 + 190 + 250 + 230 + 240 + 200 + 185 + 190 + 225 + 265}{10} = \frac{2150}{10} = 215$$ This matches the given mean. 3. **Median calculation:** The median is the middle value when data is ordered. The ordered data is: $$175, 185, 190, 190, 200, 225, 230, 240, 250, 265$$ Since there are 10 values (even), median is average of 5th and 6th values: $$m = \frac{200 + 225}{2} = \frac{425}{2} = 212.5$$ The user wrote $222.5$ which is incorrect; correct median is $212.5$. 4. **Range calculation:** Range is max minus min: $$265 - 175 = 90$$ This matches the given range. 5. **Variance calculation:** Variance formula for sample variance is: $$s^2 = \frac{\sum (x_i - \bar{X})^2}{n-1}$$ Calculate each squared difference: $$(175-215)^2 = 1600$$ $$(185-215)^2 = 900$$ $$(190-215)^2 = 625$$ $$(190-215)^2 = 625$$ $$(200-215)^2 = 225$$ $$(225-215)^2 = 100$$ $$(230-215)^2 = 225$$ $$(240-215)^2 = 625$$ $$(250-215)^2 = 1225$$ $$(265-215)^2 = 2500$$ Sum these: $$1600 + 900 + 625 + 625 + 225 + 100 + 225 + 625 + 1225 + 2500 = 8650$$ Divide by $n-1=9$: $$s^2 = \frac{8650}{9} = 961.11$$ This matches the given variance. 6. **Standard deviation calculation:** Standard deviation is square root of variance: $$s = \sqrt{961.11} = 31.00$$ This matches the given standard deviation. **Summary:** Mean, range, variance, and standard deviation calculations are correct. Median calculation is incorrect; correct median is $212.5$ not $222.5$.