Standard deviation

Grade 9 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   A set is 4, 6, 8, 10, 12, with mean 8 and SD about 2.8. Esme adds a sixth value, 8. What happens to the SD?

   • It rises, because there are now more values in the set to spread out.
   • It falls to about 2.4, because 5 of the 6 values are unchanged and one is added.
   • It falls to about 2.6: the new value adds no gap, yet six values share the total.
   • It stays at 2.8, because the new value equals the mean and so cannot change anything.
2. 2.

   A set is 10, 11, 12, 13, 14, with mean 12 and SD about 1.4. Kira adds a sixth value, 30. What happens to the SD?

   • It jumps to about 6.8: the far value's squared gap, 225, is only one value yet dominates.
   • It falls, because six values give a steadier picture than five values did, and 30 is just one.
   • It rises to about 5.0, the average of the six gaps from the new mean.
   • It stays near 1.4, because 30 is only one of the six values and the others have not moved.
3. 3.

   A set is 2, 10, 11, 13, 14, 22, with mean 12 and SD about 5.9. Luz removes the smallest and largest values. What happens to the SD?

   • It stays about 5.9, because the mean is still 12 after the two values are removed.
   • It falls to about 3.9, because 4 of the 6 values remain, two thirds of the set.
   • It falls sharply, to about 1.6: the two largest gaps, 10 and 10, are gone.
   • It rises, because fewer values are left to share the squared gaps among them.
4. 4.

   A set is 8, 9, 10, 11, 12, with mean 10 and SD about 1.4. Jun adds eleven more values, all equal to 10. What happens to the SD?

   • It falls to about 0.8: the squared gaps total 10, shared by all 16 values.
   • It rises, because there are many more values to be spread out than before.
   • It stays at 1.4, because every one of the eleven new values equals the mean.
   • It falls to 0, because most of the values in the set now equal the mean.
5. 5.

   Mean 471, SD 583. Median 240, IQR 25 (Q1 225, Q3 250). Range 1,690. Zee lists 7 recent home sales, in $1,000s: 210, 225, 230, 240, 245, 250, 1,900. Which pair of summaries best shows a typical sale and the spread?

   • Median 240 and IQR 25: the 1,900 sale drags the mean and SD far above all other sales.
   • Median 240 and SD 583, because the median is typical and the SD uses every price.
   • Range 1,690, because it shows how far apart the prices are.
   • Mean 471 and SD 583, because they use every price and so describe the data most fairly.
6. 6.

   Mean 17, SD 2.8. Median 17, IQR 4 (Q1 15, Q3 19). Range 10. Pia's 10 quiz scores are 12, 14, 15, 16, 17, 17, 18, 19, 20, 22. Which pair of summaries best shows a typical score and the spread?

   • Mean 17 and SD 2.8: the scores balance around 17, with none far-off.
   • Range 10, because it shows the lowest and highest scores.
   • Mean 17 and IQR 4, because the mean uses every score and the IQR is simple.
   • Median 17 and IQR 4, because the median is a safer choice than the mean.
7. 7.

   Mean 25.9, SD 14.0. Median 21, IQR 3 (Q1 19, Q3 22). Range 42. Ren logs a runner's weekly distance in km: 18, 19, 20, 21, 21, 22, 60 (a race week). Which pair of summaries best shows a normal week?

   • Median 21 and SD 14.0, because the median is typical and the SD uses every single week.
   • Range 42, because it shows how far the weeks differ.
   • Mean 25.9 and SD 14.0, because they use every week and so are the most accurate summaries.
   • Median 21 and IQR 3: the race week of 60 inflates the mean and SD, hiding all normal weeks.
8. 8.

   A random sample of 30 of 600 students has sleep mean 7.4 h and SD 0.9 h. Bo hears a claim: all 600 sleep 6.5 to 8.3 h. Best verdict?

   • Supported: 30 random students show exactly what all 600 do, so the claim holds for everyone.
   • Supported: every value lies within one SD of the mean, 7.4 ± 0.9, so all 600 students fit the claim.
   • Not supported: an SD of 0.9 means students differ by only 0.9 hours, so the claim is too wide.
   • Not supported: an SD is a typical distance, not a limit, and only 30 students are a sample.
9. 9.

   A random sample of 25 rides has wait mean 9 min and SD 2 min. Ari hears a claim: no ride will ever wait more than 11 min. Best verdict?

   • Supported: 25 random rides show exactly what every ride does, so the claim holds for all rides.
   • Not supported: an SD is a typical distance, not a limit, and only 25 rides are a sample.
   • Supported: every value lies within one SD of the mean, 9 ± 2, so no ride waits past 11.
   • Not supported: an SD of 2 means waits differ by only 2 minutes, so the claim is too wide.
10. 10.

    All 22 students in one class took a quiz: mean 80, SD 6. Noah hears a claim: every student scored from 74 to 86. Best verdict?

    • Supported: every value lies within one SD of the mean, 80 ± 6, so all 22 scores fit the band.
    • Not supported: an SD of 6 means scores differ by only 6 points, so the claim is too wide.
    • Supported: one class shows exactly what every class does, so the claim holds for all classes.
    • Not supported: an SD is a typical distance, not a limit, so not all 22 scored from 74 to 86.