1.3.9 practice

AP Statistics Assignment: Standard Deviation and Variance Pag e 1 of 6 Remember t h at th e s tandard devi ation and variance can onl y be used when t h e mean i s u sed as t h e m easur e of cent er. Th e v ari ance i s th e average of t h e squares o f th e devi ation s o f th e o bservation s from t hei r mean. T herefore, t h e v ari ance doesn 't h ave t h e same u nit of measu rement as th e o rigin al data. H ow ever, si n ce th e s tandard devi ation is th e square r oot of t h e v ari ance, i t does h ave th e same u nit of measu rement as th e o rigin al data. T hi s i s wh y w e o ften prefer t alkin g about t h e s tandard devi ation as oppo sed t o t h e v ariation . The s tandard devi ation can be thought of as "t h e average di st ance o f each dat a po int f ro m it s mean." Al thou gh thi s i s somewhat i mpreci se, i t 's a go o d way of thin k about t h e s tandard devi ation . Be su re t h at you k now when t o use th e sample standard devi ation f ormul a ( s ) and t h e population s tandard devi ation f ormul a ( σ ). T hi s tabl e may h elp: Situation Sample Std. Deviation Formula ( s ) Population Std. Deviation Formula ( σ ) Calculatin g st d. devi ation wi th all members o f th e po pulation x Estimatin g po pulation standard devi ation u sin g sample x Remember t h at th e sample standard devi ation f ormul a do es not gi ve you t h e s tandard deviation of your sampl e; i t estimates t h e s tandard devi ation of t h e po pulation . Fo r example, su ppo se you have t h e f ollowin g 5 t est s co res out of a po pulation of 30: {87 , 92, 74, 70, 66 } . Th e sample standard devi ation 2 2 2 2 2 2 ( x − x ) + ( x − x ) + ... + ( x − x ) (87 − 77.8) + (92 − 77.8) + ... + (66 − 77.8) 1 2 n s = = = 11.189 n − 1 4 i s th e s tatisti c th at you'd use to es timat e t h e s tandard devi ation of t h e en ti re population of 30 test s co res. The s tandard devi ation of t h e o bservation s et { 87 , 92, 74 , 70, 66 }, 2 2 2 ( x − μ ) + ( x − μ ) + ... + ( x − μ ) 1 2 30 σ = 30 woul d give you t h e s tandard devi ation of al l 30 dat a po int s in t h e po pulation . To avo i d ambigu it y when y ou 're co mmunicatin g w ith others, do n 't call a data s et a sample unl ess you i nten d t o use it t o i n fer somethin g ab out t h e po pulation f ro m which i t 's dra wn. Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.

AP Statistics Pag e 2 of 6 Assignment: Standard Deviation and Variance Fo r each c alculation , use th e s ymbol f o r t h e m easur e y ou are c alculating. For example, if you 're ca lculatin g a samp l e m ean, use th e symbol x when ref errin g t o y ou r an swer. When ref errin g t o t h e v ari ance, use th e s ymbol s s 2 and σ 2 . Don 't forget t o i nclu de u nit s in y ou r an swers. 1. Do not use your calculator for this one. Write out the formulas and show your work. Round your answers to 3 decimal places. (You'll be able to use your calculator on the problems after this one.) (4 points) You 're giv en a sample with n = 8 measurement s: 3, 1, 5, 6, 4, 4, 3, 5. A. Calcul ate th e range (maxi mu m valu e ­ minimu m value). B. Calcul ate th e sample mean. C. Calcul ate th e sample vari ance and sample standard devi ation. D. Th e range is approximatel y how m any sample standard devi ations? Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. Therangeiss ca its 5 3.875 estitsta 4 3 5 8 3.875 o a inando less The range is about standarddeviations

AP Statistics Pag e 4 of 6 Assignment: Standard Deviation and Variance 3. Consider the following set of data, which represents a simple random sample of heights (in inches) of full-grown corn plants in Horseyetown, WI: (78.5, 88.8, 76.2, 73.0, 79.5, 66.9, 72.4, 86.6, 67.6, 73.5, 72.6, 72.6, 80.0, 69.1, 79.1, 73.9, 72.3, 83.3, 64.3, 69.1) Round your answers to 2 decimal places. (4 points) A. Calcul ate th e ran ge. B. Calcul ate th e sample mean. C. Calcul ate sampl e s tatisti cs to es timat e t h e po pulation s tandard devi ation an d vari ance o f all f ull-grown c orn pl ant s in H orseyetown , W I. 4. In Goateyeville, MN, they use a new type of fish fertilizer. A random sampling of the heights of 20 full-grown corn plants in Goateyeville yields the following data: (heights are in inches) (41.0, 51.1, 65.8, 57.3, 35.8, 61.5, 62.5, 65.8, 72.9, 62.7, 76.4, 81.9, 77.7, 79.0, 73.2, 52.1, 72.0, 69.7, 76.4, 73.4) Round your answers to 2 decimal places. (4 points) A. Calcul ate th e ran ge. B. Calcul ate th e sample mean. C. Calcul ate sampl e s tatisti cs to es timat e t h e po pulation s tandard devi ation an d vari ance o f all f ull-grown c orn pl ant s in G oateyeville, MN . Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. range aia inches 4 aus inches o osa inches and o 42.4s inches range as inches E esur inches a ice inches and nois inches

AP Statistics Pag e 5 of 6 Assignment: Standard Deviation and Variance 5. Draw histograms of the distributions from Questions 2 and 4. (4 points) A. How wo ul d yo u c haracteri ze t hei r s hapes? B. How do t hei r means compare? C. How do t hei r s tandard devi ation s compare? D. How do skew and range affect t h e s tandard devi ation ; does on e aff ect i t m o re t h an the other? Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. The data from question aand4 are aft skewed Their means are the same The standarddeviation in question 4is much larger skewand range affects the standard deviation The range in question a is smaller butinthis case it does not have a large effect on the data presented