An environmental science class recorded the daily PM2.5 air quality index (AQI) values for… — SAT

An environmental science class recorded the daily PM2.5 air quality index (AQI) values for a city over a 7-day period. The recorded values for 6 of the days were $38$ , $42$ , $45$ , $49$ , $52$ , and $58$ . The AQI value for the 7th day, $x$ , is unknown. If the mean AQI value for the 7 days is $48$ , what is the median AQI value for the 7 days?

Answer: 49

Answer

The median AQI value for the 7 days is 49.

The correct median is 49. Multiplying the mean of 48 by the 7 days yields a total sum of 336. Subtracting the sum of the 6 known days (284) gives the 7th day's value as 52. Ordering all 7 values from least to greatest (

38, 42, 45, 49, 52, 52, 58

) reveals that the 4th value (the median) is 49.

Step-by-Step Solution

Calculate the sum of all 7 AQI values using the given mean.

The total sum is

7 \times 48 = 336

The mean of a dataset is the sum of its values divided by the number of values, so the sum is the mean multiplied by the number of values.

Calculate the sum of the 6 known AQI values.

38 + 42 + 45 + 49 + 52 + 58 = 284

This determines the total contribution of the known days to the sum.

Find the 7th AQI value (

x

x = 336 - 284 = 52

Subtracting the sum of the 6 known values from the total sum gives the value of the 7th day.

Sort the complete list of 7 values in ascending order.

38, 42, 45, 49, 52, 52, 58

Finding the median requires arranging the data from least to greatest.

Identify the median.

The median is

49

For an odd number of values (7), the median is the middle value in the sorted list (the 4th value).

Key Concept

Calculating and comparing measures of center (mean and median) for a data distribution

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