Background | Enter Data | Analyze Data | Interpret Data | Report Data Z-Scores Sometimes we want to do more than summarize a bunch of scores. Sometimes we want to talk about particular scores within the bunch. We may want to tell other people about whether or not a score is above or below average. We may want to tell other people how far away a particular score is from average. We might also want to compare scores from different bunches of data. We will want to know which score is better. Z- scores can help with all of this. They Tell Us Important Things Z-Scores tell us whether a particular score is equal to the mean, below the mean or above the mean of a bunch of scores. They can also tell us how far a particular score is away from the mean. Is a particular score close to the mean or far away? If a Z-Score…. ü Has a value of 0, it is equal to the group mean. ü Is positive, it is above the group mean.
Transcript
Background | Enter Data | Analyze Data | Interpret Data | Report Data
Z-Scores Sometimes we want to do more than summarize a bunch of scores. Sometimes we want to talk about particular scores within the bunch. We may want to tell other people about whether or not a score is above or below average. We may want to tell other people how far away a particular score is from average. We might also want to compare scores from different bunches of data. We will want to know which score is better. Z-scores can help with all of this.
They Tell Us Important Things Z-Scores tell us whether a particular score is equal to the mean, below the mean or above the mean of a bunch of scores. They can also tell us how far a particular score is away from the mean. Is a particular score close to the mean or far away?
If a Z-Score…. ü Has a value of 0, it is equal to the group mean. ü Is positive, it is above the group mean. ü Is negative, it is below the group mean. ü Is equal to +1, it is 1 Standard Deviation above the mean. ü Is equal to +2, it is 2 Standard Deviations above the mean. ü Is equal to -1, it is 1 Standard Deviation below the mean. ü Is equal to -2, it is 2 Standard Deviations below the mean.
Z-Scores Can Help Us Understand… How typical a particular score is within bunch of scores. If data are normally distributed, approximately 95% of the data should have Z-score between -2 and +2. Z-scores that do not fall within this range may be less typical of the data in a bunch of scores.
Individual scores from different bunches of data. We can use Z-scores to standardize scores from different groups of data. Then we can compare raw scores from different bunches of data.
How Do You Calculate a Z-Score/ Sigma Level?Jeff Sauro • June 14, 2004
The benefit of using a z-score in usability metrics was explained in "What's a
Z-Score and why use it in Usability Testing?" this article discusses different
ways of calculating a z-score.
The short answer is: It depends on your data and what you're looking for. If
you've encountered the z-score in a statistics book you usually get some
formula like:
The above formula is for obtaining a z-score for an entire population. Usability
testing obviously samples a very small subset of the population and thus the
following formula is used:
Where x-bar and s are used as estimators for the population's true mean and
standard deviation. Both formulas essentially calculate the same thing:
Calculating a Z-Score Example
For example, lets say you took the GRE a few weeks ago and got
scores of 630 Verbal and 700 Quantitative. How good are these
completes a task, but it does matter if a user takes too long. For example, say
you and your product team determined that a task should be completed in 120
seconds. 120 seconds becomes your Upper Spec Limit (USL). You sampled 10
users and got these task times:
Sample
100
99
101
125
100
123
96
90
98
116
USL: 120
Mean: 104
StDev: 12
To calculate the process sigma you subtract the mean (104) of
the sample from the target (120) and divide by the sample
standard deviation (12). For Sample 1 the process sigma is -
1.32σ. The visual representation of the data can be seen below:
In the case of task times, a negative process sigma is ideal--as you want more people completing the task below the task time, not above it. You can simply drop the negative when communicating the results in the event it causes confusion. If you were to make radical improvements to the UI and then sampled another set of ten users, here are more results:
In the redesign, the average of the new sample is well below the spec limit and the process sigma is now very high. The corresponding defect area is now only .01% and the quality area is 99.98
Of course having users perform that much below the spec limit is not very common due to the inherent variability in user performance.
If you need more help with z-scores, see the Crash course in Z-scores, a tutorial with plenty of pictures, examples and review questions for you to grasp this concept
The z-score
The Standard Normal Distribution
Definition of the Standard Normal Distribution The Standard Normal distribution follows a normal distribution and has mean 0 and standard deviation 1
Notice that the distribution is perfectly symmetric about 0.
If a distribution is normal but not standard, we can convert a value to the Standard normal distribution table by first by finding how many standard deviations away the number is from the mean.
The z-score
The number of standard deviations from the mean is called the z-score and can be found by the formula
x - z =
Example
Find the z-score corresponding to a raw score of 132 from a normal distribution with mean 100 and standard deviation 15.
Solution
We compute
132 - z = = 2.133 15
Example
A z-score of 1.7 was found from an observation coming from a normal distribution with mean 14 and standard deviation 3. Find the raw score.
Solution
We have
x - 1.7 = 3
To solve this we just multiply both sides by the denominator 3,
(1.7)(3) = x - 14
5.1 = x - 14
x = 19.1
The z-score and Area
Often we want to find the probability that a z-score will be less than a given value, greater than a given value, or in between two values. To accomplish this, we use the table from the textbook and a few properties about the normal
We use the table. Notice the picture on the table has shaded region corresponding to the area to the left (below) a z-score. This is exactly what we want. Below are a few lines of the table.
The columns corresponds to the ones and tenths digits of the z-score and the rows correspond to the hundredths digits. For our problem we want the row 2.3 (from 2.37) and the row .07 (from 2.37). The number in the table that matches this is .9911.
In this case, we want the area to the right of 1.82. This is not what is given in the table. We can use the identity
P(z > 1.82) = 1 - P(z < 1.82)
reading the table gives
P(z < 1.82) = .9656
Our answer is
P(z > 1.82) = 1 - .9656 = .0344
Example
Find
P(-1.18 < z < 2.1)
Solution
Once again, the table does not exactly handle this type of area. However, the area between -1.18 and 2.1 is equal to the area to the left of 2.1 minus the area to the left of -1.18. That is
P(-1.18 < z < 2.1) = P(z < 2.1) - P(z < -1.18)
To find P(z < 2.1) we rewrite it as P(z < 2.10) and use the table to get
P(z < 2.10) = .9821.
The table also tells us that
P(z < -1.18) = .1190
Now subtract to get
P(-1.18 < z < 2.1) = .9821 - .1190 = .8631
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e-mail Questions and Suggestions
← scoring: Use the table below to determine your BMI rating. The table shows the World Health Organization BMI classification system. The rating scale is the same for males and females. You can also use the reverse lookup BMI table for determining your ideal weight based on height.
Fitness TestingFitness Testing > Tests > Anthropometry > Body Composition > Waist to Hip
Ratio
Waist to Hip Ratio (WHR)
←
aim: the purpose of this test to determine the ratio of waist circumference to the hip circumference, as this has been shown to be related to the risk of coronary heart disease.
← equipment required: tape measure
← procedure: A simple calculation of the measurements of the waist girth divided by the hip girth. Waist to Hip Ratio (WHR) = Gw / Gh, where Gw =
Total chronic malnutrition/Total stunted (moderate + severe)
<-2 Z score <90%
Cut off points for UnderweightUnderweight based on Weight-for-Age in z-scores and percentage of the median
Table 8: Cut off points for Underweight
Description of Nutritional Status
Weight for Age Index Z scores
Weight for Age % of median
Severe Underweight <-3 Z scores <70% Moderately Underweight ≥ - 3.0 but <-2.0 ≥ 70% and <80%Total Underweight (moderate plus severe)
<-2 Z score <80%
Normal ≥-2 Z-scores ≥ 80%
Using a global classifications of malnutritionThe following classifications for malnutrition have been established by WHO as levels for interpretingWFH, HFA and WFA z-scores (WHO 2002).
For acute malnutrition (wasting), care needs to be taken to assess the context; a prevalence classified as “poor/medium” but which is likely to get worst will have different programmatic implications than a prevalence classified as “serious/high” but where the situation is likely to improve (e.g. impending good harvest).
Table 9: prevalence of malnutrition and interpretation levels
Risk of mortality using MUACFor children taller/greater than 65 cm
Table 10: MUAC cut-offs and risk of mortality
Nutritional Status MUAC
Severe <11.0 cmModerate >11.0 and 12.5 cmMild malnutrition >12.5 and 13.5cmSatisfactory nutritional status > 13.5cmNote: New WHO standards recommend MUAC < 115 mm as criteria for severe malnutrition amongchildren of age 6 months and above.
6.1. NCHS/WHO Reference StandardsThe reference standards most commonly usedto standardize measurements were developedby the US National Center for Health Statistics(NCHS) and are recommended for internationaluse by the World Health Organization. Thereference population chosen by NCHS wasa statistically valid random population ofhealthy infants and children. Questions havefrequently been raised about the validity ofthe US-based NCHS reference standards forpopulations from other ethnic backgrounds.Available evidence suggests that until the ageof approximately 10 years, children from wellnourishedand healthy families throughoutthe world grow at approximately the samerate and attain the same height and weightas children from industrialized countries.The NCHS/WHO reference standards areavailable for children up to 18 years oldbut are most accurate when limited to usewith children up to the age of 10 years. TheNCHS/WHO international reference tablescan be used for standardizing anthropometricdata from around the world and can be foundon FANTA’s website at www.fantaproject.org/publications/anthropometry.shtml.6.2. Comparisons to the Reference Standard
References are used to standardize a child’smeasurement by comparing the child’smeasurement with the median or averagemeasure for children at the same age and sex.For example, if the length of a 3 month oldboy is 57 cm, it would be difficult to knowif that was reflective of a healthy 3 monthold boy without comparison to a referencestandard. The reference or median length fora population of 3 month old boys is 61.1 cmand the simple comparison of lengths wouldconclude that the child was almost 4 cmshorter than could be expected.When describing the differences from thereference, a numeric value can be standardizedto enable children of different ages and sexesto be compared. Using the example above,the boy is 4 cm shorter than the referencechild but this does not take the age or the sexof the child into consideration. Comparinga 4 cm difference from the reference for a6. Comparison of Anthropometric Data to Reference Standards40child 3 months old is not the same as a 4 cmdifference from the reference for a 9 year oldchild, because of their relatively different bodysizes.Taking age and sex into consideration,differences in measurements can be expresseda number of ways:• standard deviation units, or Z-scores• percentage of the median• percentilesTo standardize reporting, USAID recommendsthat Cooperating Sponsors calculatepercentages of children below cut-offs as well
as other statistics using Z-scores. If Z-scorescannot be used, percentage of the medianshould be used.6.3. Standard Deviation Units or Z-ScoresZ-scores are more commonly used by theinternational nutrition community becausethey offer two major advantages. First, usingZ-scores allows us to identify a fixed pointin the distributions of different indices andacross different ages. For all indices for allages, 2.28% of the reference population liebelow a cut-off of -2 Z-scores. The percent ofthe median does not have this characteristic.For example, because weight and heighthave different distributions (variances), -2Z-scores on the weight-for-age distributionis about 80% of the median, and -2 Z-scoreson the height-for-age distribution is about90% of the median. Further, the proportionof the population identified by a particularpercentage of the median varies at differentages on the same index.The second major advantage of using Zscoresis that useful summary statistics can becalculated from them. The approach allows themean and standard deviation to be calculatedfor the Z-scores for a group of children. TheZ-score application is considered the simplestway of describing the reference population andmaking comparisons to it. It is the statisticrecommended for use when reporting results ofnutritional assessments. Examples of Z-scorecalculations are presented in Appendix 1.The Z-score or standard deviation unit (SD)is defined as the difference between the valuefor an individual and the median value ofthe reference population for the same age or
height, divided by the standard deviation of thereference population. This can be written inequation form as:6.4. Percentage of the Median and PercentilesThe percentage of the median is defined as theratio of a measured or observed value in theindividual to the median value of the referencedata for the same age or height for the specificsex, expressed as a percentage. This can bewritten in equation form as:(observed value) - (median reference value)standard deviation of reference populationZ-score (or SD-score) =observed valuemedian value of reference populationPercent of median = x 1006. Comparison of Anthropometric Data to Reference Standards41The median is the value at exactly the midpointbetween the largest and smallest. If achild’s measurement is exactly the same asthe median of the reference population we saythat they are “100% of the median.” Examplesof calculations for percent of median can befound in Appendix 1.The percentile is the rank position of anindividual on a given reference distribution,stated in terms of what percentage of the groupthe individual equals or exceeds. Percentileswill not be presented in this guide.The distribution of Z-scores follows a normal(bell-shaped or Gaussian) distribution. Thecommonly used cut-offs of -3, -2, and -1 Zscoresare, respectively, the 0.13th, 2.28th,and 15.8th percentiles. The percentiles can bethought of as the percentage of children in the
reference population below the equivalent cutoff.Approximately 0.13 percent of childrenwould be expected to be below -3 Z-score in anormally distributed population.Z-score Percentile-3 0.13-2 2.28-1 15.86.5. Cut-offsThe use of a cut-off enables the differentindividual measurements to be converted intoprevalence statistics. Cut-offs are also usedfor identifying those children suffering fromor at a higher risk of adverse outcomes. Thechildren screened under such circumstancesmay be identified as eligible for special care.The most commonly-used cut-off withZ-scores is -2 standard deviations,irrespective of the indicator used. Thismeans children with a Z-score forunderweight, stunting or wasting, below-2 SD are considered moderately orseverely malnourished. For example, achild with a Z-score for height-for-age of-2.56 is considered stunted, whereas a childwith a Z-score of -1.78 is not classified asstunted.In the reference population, by definition,2.28% of the children would be below -2 SDand 0.13% would be below -3 SD (a cut-offreflective of a severe condition). In somecases, the cut-off for defining malnutritionused is -1 SD (e.g. in Latin America). In thereference or healthy population, 15.8% wouldbe below a cut-off of -1 SD. The use of -1 SDis generally discouraged as a cut-off due to thelarge percentage of healthy children normally
falling below this cut-off. For example, the1995 DHS survey using a –2 SD cut-off forstunting in Uganda found a 36% prevalence ofstunting in under-three year olds. This levelof stunting is about 16 times the level of thereference population.A comparison of cutoffs for percent of medianand Z-scores illustrates the following:90% = -1 Z-score80% = -2 Z-score70% = -3 Z-score (approx.)60% = -4 Z-score (approx.)6.5.1. Cut-off points for MUAC for the6 - 59 month age groupMUAC cut-offs are somewhat arbitrarydue to its lack of precision as a measure ofmalnutrition. A cut-off of 11.0 cm can beused for screening severely malnourishedchildren. Those children with MUAC below12.5 cm with or without edema are classifiedas moderate and severe.Global Acute Malnutrition is a termgenerally used in emergency settings. Theglobal malnutrition rate refers to the percentof children 6 to 59 months with weight-forheightbelow -2 Z-scores or 80% medianor MUAC below 12.5 cm, with or withoutedema. This refers to all moderate and severemalnutrition combined. The combination ofa low weight-for-height and any child withedema contributes to those children counted asin the global acute malnutrition statistic.C O M PA R I S O N O F A N T H RO P O M E T R I C DATA TO R E F E R E N C E S TA N DA R D S PA RT 6 .6. Comparison of Anthropometric Data to Reference Standards42
6.5.2. Malnutrition ClassificationSystemsThe cut-off points for different malnutritionclassification systems are listed below. Themost widely used system is WHO classification(Z-scores). The Road-to-Health (RTH) systemis typically seen in clinic-based growthmonitoringsystems. The Gomez systemwas widely used in the 1960s and 1970s,but is only used in a few countries now. Ananalysis of prevalence elicits different resultsfrom different systems. These results wouldnot be directly comparable. The differenceis especially broad at the severe malnutritioncut-off between the WHO method (Z-scores)and percent of median methods. At 60% ofthe median, the closest corresponding Z-scoreis –4. The WHO method is recommended foranalysis and presentation of data (see Part6.2).Mild, moderate and severe are differentin each of the classification systems listedbelow. It is important to use the same systemto analyze and present data. The RTH andGomez classification systems typically useweight-for-age.System Cut-off Malnutrition classificationWHO < -1 to > -2 Z-score mild< -2 to > -3 Z-score moderate< -3 Z-score severeRTH > 80% of median normal60% - < 80% of median mild-to-moderate< 60% of median severeGomez > 90% of median normal75% - < 90% of median mild60% - < 75% of median moderate< 60% of median severe
___
6. ComparisoOur girl therefore has moderate protein-energy malnutrition, as defined by weight-for-height z-score.
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