What does "a value that lies z or more standard deviations above the expected value" mean?

This phrase came from the book Understanding Probability written by Henk Tijms. The original sentence is:

the probability that a normally distributed random variable will take on a value that lies z or more standard deviations above the expected value is equal to 1 − Φ(z) for z > 0, as is the probability of a value that lies z or more standard deviations below the expected value.

I don't understand what lies means in this context. I have checked The Oxford Advanced Learner's Dictionary, and Webster's online; none of their explanations refer lie as a transitive verb. How can it used with z or more standard deviations, which is obviously a noun phrase?

Could anyone explain to me:

1. What does the sentence mean?
2. The grammar underlying this sentence? Am I right that the lie used here is actually a transitive verb?
• I have added the lower case phi to your quotation. If φ is not the correct phi, please copy and paste the appropriate version in: ϕ Φ . If you wish to find special characters not on your keyboard in the future, you can use the program character map (Windows and Ubuntu have this; I don't know about Macs, other Linux distros or mobile devices) or search for the name of the character, then copy and paste. Commented Aug 20, 2014 at 11:15
• It's the uppercase one, I reedit the question, thank you.
– Rui
Commented Aug 20, 2014 at 13:25

In this case, z or more standard deviations is a measure of distance. You're right about its composition but not about its usage; it's a noun phrase but lies is still intransitive. The phrase quantifies above the expected value rather than functioning as the object of lies. The entire clause z or more standard deviations above the expected value acts as an adverb to modify lies; it tells us precisely where the value lies.

Here's a break down and subsequent simplification.

[A value that]1 [lies]2 [z or more standard deviations]3 [above the expected value]4.

1. The subject; let's simplify by replacing this with the city.
2. The verb.
3. How far; let's simplify by replacing this with 50 kilometres.
4. Where; let's simplify by replacing this with away.

Now, the sentence becomes:

The city lies 50 kilometres away.

The grammar is exactly the same. 50 kilometres away is an adverbial clause telling us where the city lies.

Of the Oxford Advanced Learner's Dictionary's definitions, the fourth one is most applicable:

4 [intransitive]
(of a town, natural feature, etc.) to be located in a particular place
The town lies on the coast.

When the values and distribution are plotted, such as on a histogram, a value can be said to lie somewhere on the plot. The use of lie implies some kind of visualization.

You can parse the sentence as

• Subject: The probability
• Subordinate clause: that
• Subject: a normally distributed random variable
• Verb: will take on
• Object: a value
• Subordinate clause: that
• Verb: lies
• Adverbial phrase: z or more standard deviations
• Prepositional phrase: above the expected value
• Verb: is
• Prepositional phrase: to 1 − φ(z)
• Prepositional phrase: for z > 0

The verb lie is intransitive: its complement is a prepositional phrase ("above the expected value").

From a technical viewpoint, the sentence is much more complex than it needs to be. I would probably rephrase it as

The probability that a normally distributed random variable will take on a value ≥ z, where z is the number of standard deviations above the expected value, is 1 − φ(z). By symmetry, the probability of a value z or more standard deviations below the expected value is also 1 − φ(z).

• I realize I'm nitpicking semantics, but your amended version changes the meaning of the calculation with the statement random variable will take on a value ≥ z (only in the first sentence; the second matches the original's meaning). The random variable is a data point and not bounded by or compared to z, which is the number of standard deviations. Consider the simple case with expected value = -1000, z = 1, φ(1) = 0.5, standard deviation = 2. P(RV ≥ 1) [your statement] is effectively zero; P(RV ≥ -998) [the original] = 0.5. Commented Aug 20, 2014 at 12:10
• Thank you, It's really difficult to decide which answer to accept, but I had an vague understanding on the sentence already, and Esoteric's answer is more about the grammar, so I'll accept his answer.
– Rui
Commented Aug 20, 2014 at 13:43