Final answer:
After subtracting 7 from each value of the original data set, the mean is 37, the median is 35.5, there is no mode, the range is 24, and the standard deviation will remain the same as that of the original data set (the calculation of which typically requires a calculator or statistical software).
Step-by-step explanation:
To find the mean, median, mode, range, and standard deviation of the given data set after adding -7 to each value, we will first adjust the given data set: 56, 37, 41, 50, 38, 44, 32, 54. By subtracting 7 from each value, we get the new data set: 49, 30, 34, 43, 31, 37, 25, 47.
Next, we calculate each of the statistical measures for the adjusted data set:
- Mean (average): sum all the numbers and divide by the number of values. (49 + 30 + 34 + 43 + 31 + 37 + 25 + 47) / 8 = 296 / 8 = 37.
- Median: arrange the data in ascending order (25, 30, 31, 34, 37, 43, 47, 49) and find the middle value; since there are an even number of data points, the median is the average of the two middle values, which is (34 + 37) / 2 = 35.5.
- Mode: value(s) that appear most frequently; in this case, no number repeats, so there is no mode.
- Range: difference between the highest and lowest values, which is 49 - 25 = 24.
- To find the standard deviation, we calculate the variance (mean of the squared differences from the mean) and then take the square root of the variance. This calculation is more complex and often done with a calculator or statistical software.
By adding or subtracting a constant from each value in a data set, the measures will shift accordingly, but the spread or variability (standard deviation) remains unchanged.