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 falls to about 2.6: the new value adds no gap, yet six values share the total.
   • It rises, because there are now more values in the set to spread out.
   • It stays at 2.8, because the new value equals the mean and so cannot change anything.
   • It falls to about 2.4, because 5 of the 6 values are unchanged and one is added.
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 falls, because six values give a steadier picture than five values did, and 30 is just one.
   • It stays near 1.4, because 30 is only one of the six values and the others have not moved.
   • It jumps to about 6.8: the far value's squared gap, 225, is only one value yet dominates.
   • It rises to about 5.0, the average of the six gaps from the new mean.
3. 3.

   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 0, because most of the values in the set now equal the mean.
   • It falls to about 0.8: the squared gaps total 10, shared by all 16 values.
   • It stays at 1.4, because every one of the eleven new values equals the mean.
   • It rises, because there are many more values to be spread out than before.
4. 4.

   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?

   • Range 1,690, because it shows how far apart the prices are.
   • Median 240 and SD 583, because the median is typical and the SD uses every price.
   • Mean 471 and SD 583, because they use every price and so describe the data most fairly.
   • Median 240 and IQR 25: the 1,900 sale drags the mean and SD far above all other sales.
5. 5.

   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.
   • 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.
   • Range 10, because it shows the lowest and highest scores.
6. 6.

   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 IQR 3: the race week of 60 inflates the mean and SD, hiding all normal weeks.
   • Range 42, because it shows how far the weeks differ.
   • Median 21 and SD 14.0, because the median is typical and the SD uses every single week.
   • Mean 25.9 and SD 14.0, because they use every week and so are the most accurate summaries.
7. 7.

   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: every value lies within one SD of the mean, 7.4 ± 0.9, so all 600 students fit the claim.
   • Supported: 30 random students show exactly what all 600 do, so the claim holds for everyone.
   • Not supported: an SD is a typical distance, not a limit, and only 30 students are a sample.
   • Not supported: an SD of 0.9 means students differ by only 0.9 hours, so the claim is too wide.
8. 8.

   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?

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

   A random sample of 40 of 500 sunflowers has height mean 130 cm and SD 15 cm. Milo hears a claim: none of the 500 is under 100 cm. Best verdict?

   • Supported: 40 random sunflowers show exactly what all 500 do, so the claim holds for every plant.
   • Not supported: an SD is a typical distance, not a limit, and 40 sunflowers are only a sample.
   • Supported: values never fall more than two SDs from the mean, and 130 - 30 = 100.
   • Not supported: an SD of 15 means heights differ by only 15 cm, 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?

    • Not supported: an SD is a typical distance, not a limit, so not all 22 scored from 74 to 86.
    • 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.
    • Supported: every value lies within one SD of the mean, 80 ± 6, so all 22 scores fit the band.