A certain value has a standardized score \( =1.75 \). How many standard deviations from the mean does this value fall?

A standardized score, or z-score, indicates how many standard deviations a particular value is from the mean of a distribution. The formula for calculating a z-score is: \[ z = \frac{{x - \mu}}{{\sigma}} \] Where: - \( z \) is the standardized score, - \( x \) is the value, - \( \mu \) is the mean, - \( \sigma \) is the standard deviation. In your case, the standardized score is \( z = 1.75 \). This means that the value \( x \) is **1.75 standard deviations above the mean**. **Interpretation:** - **Positive z-score (e.g., 1.75):** The value is above the mean. - **Negative z-score:** The value is below the mean. - **Z-score of 0:** The value is exactly at the mean. So, a z-score of 1.75 indicates that the value is 1.75 standard deviations **above** the mean. **Example:** If the mean height of a group is 170 cm with a standard deviation of 10 cm, a person with a height corresponding to a z-score of 1.75 would be: \[ x = \mu + z \times \sigma = 170 \, \text{cm} + 1.75 \times 10 \, \text{cm} = 187.5 \, \text{cm} \] This person is 1.75 standard deviations taller than the average height in the group. **Summary:** A standardized score of 1.75 means the value is 1.75 standard deviations above the mean.